In Skyrim, is there a limit to the number of cycles (cycles of 5 times and leveling up) you can train with a particular trainer? Or is this a bug?

I trained maybe 4 or 5 times in a row for Speech with Ogmund in Markarth (I trained 5 times, then leveled up, and repeated this cycle four or five times in a row), and after that there is no longer a dialogue option to train me anymore. I went to another trainer (Babette, who I know trains Alchemy), and she allows me to train. However, I noticed that none of the people at the Mage's college have the option for me to train either. I thought I remembered them having the option.

Is this a bug, or is there a limit to the number of times you can train with a particular trainer?

The trainers work in a very simple way: Every level, you can train 5 levels of the skill you want. However, every trainer has a limit to the level he can train you to.

If I am correct there are three levels, common (lvl < 50), expert (lvl < 75) and master(lvl < 90).

When you pass the threshold of the trainer, you have to find another one which can in fact train you further or you have to train it the normal way.

Different trainers have different max levels for their respective skill.

For instance, one Archery trainer might only train you up to level 50, while another may train up to 75 or 90.

Check http://www.uesp.net/wiki/Skyrim:Trainers for a comprehensive list.