Depends on the bank, but in most cases yes, you can have several. I've got/had four Chase cards, five or six Citi, two Barclay, ... The only one I know of that only allows one card is Advanta.

You could also look at using a business card which allows multiple "employee" cards to keep track of things. This is probably not the best way to go since you lose some consumer protections on business cards, but would mean multiple cards and easy to separate charges yet only pay one bill.