If the memo line on a check states, “payment in full” and the check is cashed, can you be billed foradditional money?

Such a marking on the memo line of a check is not legally binding. By cashing it, your payee gave up none of their legal rights to collect the full sum owed (or allegedly owed).  The reason is that if someone owes money, they may not unilaterally change the amount owed or terms of payment by merely making a notation a check.  There is a common misconception that the memo line on a check has legal force; the fact is that it does not.  This line is strictly for informational purposes.  While it helps to identify the reason that a check was written, it does not bind the recipient.  If it was that easy to alter the terms of a payment, everyone would do it. 

However, if you paid the full amount that you should have paid, then your creditor cannot arbitrarily and after the fact increase the amount of your bill (unless there was some obvious error on it).

Bottom line, if you can't come to terms with your creditor, a court may have to decide what (if anything) is owed.  

You have the right to pay your legal debts (see below)--though only your legal acquired  obligations...unfortunately, that's about your only right, unless the debt is so old that the statute of limitations, or time to enforce it under the law, has passed. It is a common misunderstanding or myth to think that writing "payment in full" on a check's memo line itself has any legal effect. If you still owed money at that time, writing payment in full when you are still obligated does not do anything.

On the other hand, if you paid the full amount that you should have paid--i.e. the amount of the invoice--then the creditor cannot after the fact add more to the charge. So if you had received a bill which itself was marked final and the amount owed matched what you should have paid (such as under a contractor's estimate or work order), then unless there was some obvious error which is being rectified (e.g. you were supposed to have paid for materials, but a contractor forget to include it on the bill, which he is no doing), you should not be liable for additional amounts.

So the context or situation is what matters. If you paid everything you should have, you should be done; if you did not, and you truly should owe more, then having written paid in full,  or even the bill having been marked "final," does not necessarily cut off your obligation to finish paying.