About 8 years ago I recived a check and because I was diputing the amount I never paid it in. I know I probably can't pay i in now but could I go back to the institution (a University) and ask for another one?
You can ask but whether you will actually get anywhere is a different matter!
I seriously doubt that they will replace the cheque after all this time and debts are 'statute barred' after 6 years so you can't take them to court to try and get the money from them either.
Cheeky Chops is right - but if the uni acknowledged the debt sometime since 8 years ago the time starts from then. So for instance if they acknowledged it 5 years ago there is still a chance. But you would need proof.
I work for a bank and can assure you that this cheque can still be paid - assuming the drawer's account is (a) still open and (b) has sufficient funds to pay it.
You have nothing to lose by trying. In the event that it cannot be paid - perhaps if the drawer has stopped it - it will be returned to you anyway.