The reason the 1960 agreements didn't work were mainly three:
1) Many people (most?) at that time didn't believe in independence. They saw the 1960 agreements as something transitional to something else. This something else for many GCs was union with Greece, and for many TCs partition.
The lesson we should have learned from point 1: If you agree and sign something, then better believe in it and try to make it work. If you don't like it, then don't sign it in the first place. Signing an agreement with hopes that you will later manage to change it is something totally wrong.
2) The 1960 agreements were not very fair. I can understand the theory behind TCs having veto power on some matters, but what about the part that said TCs should get 30% of government jobs and such? In Cyprus were getting a job in the government is the dream of every middle class person, giving to some citizens double the chance of getting this kind of jobs is obviously something that wouldn't create very nice feelings on the rest.
The lesson we should have learned from point 2: While arrangements on the high level (e.g. ministers, parliament members etc) can be arranged on a ratio different from the population one, all "regular" citizens should be equal and there shouldn't be any special treatment because of somebody's religion and language. (of course in case that a part of the population is under represented somewhere, affirmative action can be taken.)
3) Some foreign powers didn't want a united independent Cyprus. They used Cyprus to serve their interests and they have us fighting between each other and loosing, while they were gaining on our loss.
The lesson we should have learned from point 3: We should stop seeing each other as something separate. We should think that we are all Cypriots and we want a better Cyprus for every Cypriot. We should be strong enough to resist their manipulations, and smart enough not to fall in their traps. They want to serve their interests and thats natural. It is time that all Cypriots together start to see how we will serve our interests as Cypriots.