For example say I have a $100 bill. If I rip it up into 2 halves, and try to cash in each half at a different bank location, could I gain $100?
You did not state which country you were referring to, so I'm going to assume Canada.
Obviously, the standard approach if you accidentally rip a $100 bill in half would be to tape the two halves together. Failing that, though, the Bank of Canada will still accept damaged bills. A torn bill consisting of more than three-fifths of the note is worth full value. A bill is worth half if between 40% and 60% of the bill remains intact. It is worth nothing if less than this remains intact. This information comes from this source and this source (The Great Canadian Trivia Book by Ray and Kearney).
In your example, each half would be worth $50. You would not be able to profit from this approach. Indeed, I know of no country where you could profit from this approach, for obvious reasons.
The US Bureau of Engraving & Printing has a process for submitting a claim for damaged or mutilated currency.
From their FAQ
Q: I have some currency that was damaged. My bank will not exchange it for undamaged currency. What can I do?
A: The Bureau of Engraving and Printing's Office of Currency Standards processes all requests for reimbursement for damaged United States currency. They decide the redemption value of torn or otherwise unfit currency by measuring the portions of the notes submitted. Generally, they reimburse the full face value if clearly more than one-half of the original note remains. Currency fragments measuring less than one-half are not redeemable.
Of course we still don't know the currency you are talking about, which most certainly would make a difference in the amount you could recover from one-half of a bill, if anything at all.
For the reasons that have already been given, this would most likely not work.
But let's say, that you managed to pull it off. At least for US currency, each half of the bill contains the serial number of the bill. I would imagine that the process would involve recording the serial number of bills that were reimbursed and it would likely be found out (eventually) that the bill was reimbursed twice. This would obviously not be good for you.
In the US, a bank is allowed to replace damaged currency if clearly more than half of the bill remains. If it's not clear that more than half of the bill remains, it would have to be turned in to the US Bureau of Engraving & Printing or the US Treasury, for investigation and possible reimbursement. Both of these agencies have programs for handling reimbursement of damaged currency.
Something missing from all the existing answers, IMHO, is the reason that the U.S. (and every country as far as I know/guess) requires you to present visibly more than 50% of the bill, in a single piece. If you tear a $100 bill in half and tape it back together, a conscientious bank teller should refuse to accept it. (I have not tested this hypothesis in practice.)
The reason is Sam Loyd's Get Off The Earth puzzle. If you're allowed to exchange a bill that's been torn in two and has a piece missing out of the middle — even a small piece — then you can multiply your money.
Each "bill" in the AFTER picture definitely has more than 60% of a bill present; but the middle two bills aren't in a single piece, so they are no longer legal tender.