The supposed lawyer you spoke with was likely incompetent if he/she did not mention the three year statute of limitations. Also, how would a landlord prove that a rent payment was late if you were not notified in writing at the time it occurred? The date of a deposit would be meaningless. Cash transferred by hand would have no traceability or dates associated with it. Also, how would the amount owed go up over time?
If there is no paper trail, no written requests to pay the rent, no request to pay late fees, then the landlord would have a hard time proving his case at trial.
You could test the landlords resolve by sending him a check for a few hundred bucks or more with the stipulation that "The cashing or depositing of this check #xxx for the amount of $xxx satisfies all debts between xxxx and xxxx." ...And see what happens.

