Technically, you can open multiple cards to receive more signing bonuses, but there are limitations. You won’t be able to open the same card multiple times, though you may be able to open a number of different cards. However, you eventually may get automatically declined if a card company sees that you’ve opened several recent accounts.
Opening several accounts also may not be a good idea, as hard inquiries when you apply for credit have a negative impact on your credit score. Multiple cards may also stretch your finances thin as you attempt to keep up with paying your bills on time on all of them.