There is no limit on the number of times you can apply to MBB firms over your lifetime. However, most offices have a time restriction in terms of time after being rejected before you can apply again - usually 1-1.5 years.
There are several exceptions to this rule. If you first apply for an internship, you are generally allowed to reapply for full time less than a year later. If you were on the border and didn't make it in your final round, firms will sometimes invite you to reapply in 6-12 months, or if you network with people at the firm they will be able to convince recruiting to make an exception and give you another chance.
There is no limit, but I would say that after the third/fourth time you apply with no positive outcome, then the application will not be considered anymore a top-priority. This is from experience where I know people who applied four times and fourth time was a charm (undergrad internship - no offer, undergrad full time - no offer, mba internship - no offer, mba full time - got in), but never met anyone who applied more times though technically is possible that they can apply as an experience hire after 3-5 years and get in.
hope this helps,
i would pick one role for each firm. Anymore may show you are unstructured and just fishing for any work. You should be clear where and what type of work you want to do.
If you do say apply for one role, but your skills are better suited for another area (and your application impresses), HR will spot this and make contact to suggest applying for another role (if you’re interested in that role of course ).
There are no official rules, however, I would not expect a good outcome if you apply more than 3 times. It also depends on the step when you were rejected. You may get 3 resume rejects but then get a dramatic improvement in your career and suddenly become relevant for consulting