FileManagement.psm1

function Read-Chunk {
    param([System.IO.StreamReader] $streamReader, [int] $chunkSize)

    $i = 0;
    while (($i -lt $chunkSize -and ($line = $streamReader.ReadLine()))) {
        $i++;
        Write-Output $line
    }
}

function Get-ChunkedContent {
    param($File, $ChunkSize)
    try {
        $stream = [System.IO.File]::OpenRead($File)
        $streamReader = New-Object "System.IO.StreamReader" -ArgumentList $stream

        $chunkNumber = 1
        $chunk = $null
        while ($chunk = (Read-Chunk $streamReader $ChunkSize)) {
            Write-Output @{ 
                Number = $chunkNumber;
                Content = $chunk;
            }
            $chunkNumber++
        }
    } finally {
        $streamReader.Dispose()
    }
}

gistfile1.txt

Import-Module FileManagement.psm1
Get-ChunkedContent "filename" 5000 | %{ $_.Content > "File$($_.Number).txt" }