Assassins can only level up past level 10 under certain circumstances. I believe they must become leaders of a den first. I believe to get to the highest level (Master Assassin, level 15 if I recall) you must do the second part of their den's memory. When they're ready, you'll get a notification that an assassin is "ready for more training."
Since there are a limited number of dens, many of your assassins will stop leveling at level 10. Experience is still tracked up to the Master Assassin rank, presumably so that in case you promoted them to den leader, that experience wouldn't go to waste.
I ended up having several of my 'non-den leader' assassins have more experience than my den leader assassins, which meant that I had to continue training the den leaders in order to unlock the Master Assassin/den leader memories. Let that be a lesson - if you've got to pick a new den leader, try to pick an assassin that is the closest to the highest level, so you unlock the related mission faster.