Sometimes, after a period of time (a few years later), you can do it again if still being offered. We did one of the Chase savings account offers a second time for DD after a two year waiting period. As we discussed on the Discover, theirs is one and done.
Overall, I'm not sure how many let you redo them, or restrictions between different offers at the same bank. I do know that cross-product will generally work. You can do a banking or credit card offer and then turn around and do the other. We did that with HSBC. Did a $200 checking account bonus (on a $1500 deposit), then turned around and did their credit card for $150 bonus for charging $500 in the first three months.
What I also know is that over time most banking offers have gone to requiring some level of direct deposit. Until now I've always stayed away from them, but with audreyh1's comment, I may try a few which do have direct deposit requirement and transfer the money from our brokerage account. Last year, one local bank specifically indicated that plain vanilla ACH transfers counted so we did that one, merely transferring money in/out of our primary bank.