A lot depends on what channels you want to record. Yes, it is possible to record multiple channels at once. Will you need multiple tuner cards to record 4-6 channels, most likely.
If you're looking to record multiple channels that you would normally view with a Cable Set Top Box, then you'd either need a set top box for each channel you want to record or use a "cablecard" TV tuner. For this, you would need either a Ceton InfiniTV tuner PCI-E card (four tuners) or the SiliconDust HDHomeRun Prime (three tuners) with a cablecard from your cable/satellite company.
If all you're looking to do is record Over-The-Air (OTA) digital broadcasts or unencrypted cable channels (channels you can view without a set top box), you can do so with multiple dual digital tuner cards like the Hauppauge WinTV 2250 HVR card.