Most cases that you buy for home-built 'puters have 4 internal 5" drive bays, some have 6. The real problem is not the case, but where to connect all the drives.
Here's the local configuration:
One SATA CDRW drive perched on top of the case.
4 internal IDE burners connected to motherboard IDE channels.
One external firewire burner.
Piles of burners on the shelf.
One can never have too many burners.