Using a hub for USB, Raspberry Pi should be able to record images snapped at a reasonable rate. The hub is needed because of current limits(700mA) on the RPi board. Images are better because you really cannot process too much information on a single processor.If you want video, one approach would be to get video from Camera 1 for 10 seconds and switch to Camera 2. Repeat until all have had a chance and restart from Camera 1 again. Are you planning to store information? This could be a problem with SD card i/o.
Pi can record video from a single USB or IP camera/webcam using free motion capture software like Motion or Xeoma for linux. Xeoma has added support for linking multiple IPs together so if you have a Pi at every webcam and on your network, you can control all from any Pi or computer. While Motion is free, Xeoma is technically commercial but it has a free version with some stripped down features. Still worth checking out. Both softwares can detect when changes in the picture to trigger recording. That way you don't fill up your memory card by recording continuously. If you have a PIR sensor or infrared LED ring for your cameras, you can record in the dark too. You may be able to get multiple cameras on one Pi (using a powered hub) but usually only at diminished resolutions. Multiple cameras running anything higher than 640x480 and you'll start to approach the maximum bandwidth for the USB bus regardless of what processor is driving your board. Good luck!
