diff --git a/public/Get-DbaBackupInformation.ps1 b/public/Get-DbaBackupInformation.ps1
index 828c7120351..4b6072d8d6f 100644
--- a/public/Get-DbaBackupInformation.ps1
+++ b/public/Get-DbaBackupInformation.ps1
@@ -326,6 +326,10 @@ function Get-DbaBackupInformation {
                             $Files += Get-XpDirTreeRestoreFile -Path "$f$($separator)FULL" -SqlInstance $server -NoRecurse
                             $Files += Get-XpDirTreeRestoreFile -Path "$f$($separator)DIFF" -SqlInstance $server -NoRecurse
                             $Files += Get-XpDirTreeRestoreFile -Path "$f$($separator)LOG" -SqlInstance $server -NoRecurse
+                        } elseif ($f -match "^s3://" -and [System.IO.Path]::GetExtension("$f").Length -eq 0) {
+                            # An S3 folder: T-SQL cannot list S3 objects (see Get-XpDirTreeRestoreFile), and handing the
+                            # folder to Read-DbaBackupHeader would only get it rejected for not being a file.
+                            Stop-Function -Message "S3 paths cannot be enumerated using T-SQL. Use explicit file paths or PowerShell-based enumeration for S3 storage." -Target $f -Continue
                         } else {
                             Write-Message -Level VeryVerbose -Message "File"
                             $Files += $f
@@ -349,7 +353,17 @@ function Get-DbaBackupInformation {
                 try {
                     $FileDetails = Read-DbaBackupHeader -SqlInstance $server -Path $Files -StorageCredential $StorageCredential -EnableException
                 } catch {
-                    Stop-Function -Message "Failure on $($server.Name)" -ErrorRecord $PSItem -Target $server.Name -Continue
+                    # This gives up on the paths of this process invocation, not on the command: a plain
+                    # Stop-Function sets the interrupt flag that Test-FunctionInterrupt reads at the top of this
+                    # block, which would drop every path piped in after these, and -Continue has no loop to
+                    # continue here (#10638). So throw under -EnableException, otherwise warn, and return from
+                    # this process invocation only.
+                    if ($EnableException) {
+                        Stop-Function -Message "Failure on $($server.Name)" -ErrorRecord $PSItem -Target $server.Name -EnableException $true
+                    } else {
+                        Write-Message -Level Warning -Message "Failure on $($server.Name)" -ErrorRecord $PSItem -Target $server.Name
+                    }
+                    return
                 }
             }
 
diff --git a/public/Read-DbaBackupHeader.ps1 b/public/Read-DbaBackupHeader.ps1
index 237e9225021..13bd6f9aae7 100644
--- a/public/Read-DbaBackupHeader.ps1
+++ b/public/Read-DbaBackupHeader.ps1
@@ -205,7 +205,7 @@ function Read-DbaBackupHeader {
         foreach ($p in $Path) {
             Write-Message -Level Verbose -Message "Checking: $p"
             if ([System.IO.Path]::GetExtension("$p").Length -eq 0) {
-                Stop-Function -Message "Path ("$p") should be a file, not a folder" -Category InvalidArgument
+                Stop-Function -Message "Path ($p) should be a file, not a folder" -Category InvalidArgument
                 return
             }
         }
diff --git a/tests/Get-DbaBackupInformation.Tests.ps1 b/tests/Get-DbaBackupInformation.Tests.ps1
index 4eba7ab549f..907fe20742c 100644
--- a/tests/Get-DbaBackupInformation.Tests.ps1
+++ b/tests/Get-DbaBackupInformation.Tests.ps1
@@ -267,4 +267,73 @@ Describe $CommandName -Tag IntegrationTests {
             $resultsSanLog.Count | Should -BeExactly 3
         }
     }
-}
\ No newline at end of file
+
+    Context "When a file is not a backup" {
+        BeforeAll {
+            # A text file where the instance can read it, so the header read fails on the content, not on access.
+            $notABackup = Join-Path -Path $TestConfig.Temp -ChildPath "dbatoolsci_notabackup_$(Get-Random).txt"
+            Set-Content -Path $notABackup -Value "dbatoolsci"
+        }
+
+        AfterAll {
+            Remove-Item -Path $notABackup -Force -ErrorAction SilentlyContinue
+        }
+
+        It "Warns without eating an iteration of the caller's loop" {
+            # The header-read catch used to run Stop-Function -Continue in the process block, where no loop
+            # encloses it - the continue escaped the command and consumed an iteration of this very loop, so
+            # the counter stayed at zero (#10638).
+            $loopCount = 0
+            foreach ($i in 1..3) {
+                $null = Get-DbaBackupInformation -SqlInstance $TestConfig.InstanceSingle -Path $notABackup -WarningAction SilentlyContinue
+                $loopCount++
+            }
+            $loopCount | Should -Be 3
+            ($WarnVar -join " ") | Should -BeLike "*Failure on*"
+            # The last call carried the message once, not twice.
+            @($WarnVar | Where-Object { $PSItem -like "*Failure on*" }).Count | Should -Be 1
+        }
+
+        It "Still reads the backup piped in after the file that is not one" {
+            # A plain Stop-Function used to set the command-wide interrupt flag for the unreadable file, and
+            # Test-FunctionInterrupt then dropped every path piped in after it.
+            $validBackup = (Get-ChildItem -Path $DestBackupDir -Filter "$dbname*.bak" | Select-Object -First 1).FullName
+            $results = $notABackup, $validBackup | Get-DbaBackupInformation -SqlInstance $TestConfig.InstanceSingle -WarningAction SilentlyContinue
+            # The full and the differential of the fixture can share one file, so the file may hold two backup sets.
+            ($results | Measure-Object).Count | Should -BeGreaterThan 0
+            $results.Database | Select-Object -Unique | Should -Be $dbname
+            ($WarnVar -join " ") | Should -BeLike "*Failure on*"
+        }
+
+        It "Throws for the file that is not a backup under -EnableException" {
+            $headerException = $null
+            $headerWarnings = $null
+            try {
+                Get-DbaBackupInformation -SqlInstance $TestConfig.InstanceSingle -Path $notABackup -EnableException -WarningVariable headerWarnings
+            } catch {
+                $headerException = $PSItem
+            }
+            # Stop-Function rethrows the inner RESTORE HEADERONLY error, so the exception text is the SQL error, not the message.
+            $headerException | Should -Not -BeNullOrEmpty
+            # Stop-Function writes the message once before it throws, as it does for every throw in dbatools; it is not
+            # written a second time.
+            @($headerWarnings | Where-Object { $PSItem -like "*Failure on*" }).Count | Should -Be 1
+        }
+    }
+
+    Context "When Path is an S3 folder" {
+        It "Warns that S3 folders cannot be enumerated and returns nothing" {
+            # An S3 URL skips xp_dirtree, and a folder then went to Read-DbaBackupHeader as if it were a file, which
+            # rejected it with a message about files and folders instead of the one about S3 enumeration. On top of
+            # that the escaping continue ended the CI test before its assertions, so nobody noticed.
+            $splatS3Folder = @{
+                SqlInstance   = $TestConfig.InstanceSingle
+                Path          = "s3://dbatoolsci.invalid/bucket/folder/"
+                WarningAction = "SilentlyContinue"
+            }
+            $results = Get-DbaBackupInformation @splatS3Folder
+            $results | Should -BeNullOrEmpty
+            ($WarnVar -join " ") | Should -BeLike "*S3 paths cannot be enumerated using T-SQL*"
+        }
+    }
+}
diff --git a/tests/Read-DbaBackupHeader.Tests.ps1 b/tests/Read-DbaBackupHeader.Tests.ps1
index 6f2853d5512..f9ccdc6a922 100644
--- a/tests/Read-DbaBackupHeader.Tests.ps1
+++ b/tests/Read-DbaBackupHeader.Tests.ps1
@@ -27,4 +27,21 @@ Describe $CommandName -Tag UnitTests {
     Integration test should appear below and are custom to the command you are writing.
     Read https://github.com/dataplat/dbatools/blob/development/contributing.md#tests
     for more guidence.
-#>
\ No newline at end of file
+#>
+
+Describe $CommandName -Tag IntegrationTests {
+    Context "When Path is a folder" {
+        It "Warns that it needs a file and returns nothing" {
+            # The message used to be built from a broken string literal, "Path ("$p") should be a file, not a folder",
+            # which handed $p to Stop-Function as a positional argument and turned the warning into a binding error.
+            $splatFolder = @{
+                SqlInstance   = $TestConfig.InstanceSingle
+                Path          = $TestConfig.Temp
+                WarningAction = "SilentlyContinue"
+            }
+            $results = Read-DbaBackupHeader @splatFolder
+            $results | Should -BeNullOrEmpty
+            ($WarnVar -join " ") | Should -BeLike "*should be a file, not a folder*"
+        }
+    }
+}