That said, there are still instances where the system is going to choke on a merge duplicates request.
One reason that records might merge, is that merging the records would result in some field or other going over the system limit for that field. The limits in Family Tree are pretty generous. The most common limit you'll run up against is in the Notes field. Each person record can only have a total of 20 notes. That is a good first thing to check if you are merging ancestors common to many Family Tree users (like early LDS Church members). Look at the Notes field and see if you can prune them.
Sometimes you are trying to merge with someone who has already been merged with another record or deleted. This usually happens when you are merging by ID and have not checked recently to see if the duplicate record is still in the system.
You can't merge individuals if one record shows as a female and the other as a male. You'll need to change the sex for one of them. Sometimes you can make that change yourself. If you can't, and you have good evidence of the correct gender, contact FamilySearch Support. You'll need to supply the IDs of both records and convincing evidence of the correct gender. A data admin can make the change. Be patient--they are sometimes flooded with requests.
You can't merge the record of a living person with a record of a deceased person. If you know the person is deceased, enter the death information on the living record and then you should be able to merge.
You cannot merge two records that are both for living people. Since you should only be seeing living people because you entered them into the system, you should be able to just delete the duplicate record. If you can't contact FamilySearch Support for help.
Some records are locked for one reason or another. You can't merge locked records. Again, you can contact FamilySearch Support to see if it is possible to get the records merged.
Sometimes you lose track of what you have done and try to merge records you or someone else merged in the past 24 hours. Please allow 24 hours for a merge to complete in all systems. Unfortunately, there is not an easy way to know if this is the problem. So, you could wait 24 hours with any merge that the system rejects and try again, or you can contact FamilySearch Support. Support agents have access to a tool that can tell them why the records won't merge.
You can't merge two records if those two individuals currently show as parent and child. You'll need to examine the records and make corrections to the relationships and then try again.
Sometimes you simply entered the ID incorrectly when you try a Merge by ID. If you see a "Person Not Found" message, double-check the IDs.
In any case where you can't merge and seems like you should be able to, check for problems like those above. If all else fails, contact FamilySearch Support. Make it easy on the support agent by having the IDs of the people you want to merge handy. Also know that the support agent is likely to ask to sign in as your helper so as to be able to see things the way you are seeing them. So, be prepared to supply your username (the one you use when you sign in on FamilySearch.org) and your helper number.
To find your helper number, go to FamilySearch.org and sign in.
Click your name in the upper right and click Settings in the dark box.
Don't let yourself get frustrated if you can't merge. Talk to a consultant in your ward if you are LDS; go to a Family History Center for help whether you are LDS or not; send me an email at firstname.lastname@example.org (if you don't mind if I use your problem as an illustration in a blog post); or contact the good people in Support. Problems can usually be fixed, or at least explained.