I've had this happen many times on my fender basses. Based on my experience, the cause is the string/silk getting nagged on the string retainer. When you get to a certain string tension, the string cannot slip past the retainer. You end up tightening the portion between the tuner and the retainer until the string snaps.
I have solved the problem by putting a spacer under the string retainer (along with a longer screw to compensate). This lowers the break angle over the nut and also reduces the sharp angle where the string contracts the retainer.
Others have solved the issue by smoothing the under surface on the retainer itself to reduce the catch on the string.
Many times on several different basses? That sounds a bit strange... Are you by any chance using heavier gauge strings than 45-105? Are you absolutely sure the problem isn't with the nut rather than the string tree? Maybe the strings get stuck at the nut rather than the strings tree, because the nut slots are too tight? 45-105 are usually the heaviest gauges factory made nuts will accommodate without potential issues. Maybe raising the string tree only cured the symptoms (by relieving pressure over the nut) rather than the disease?
I ask because I own 40+ basses and I work at a guitar workshop where I go through lots of basses every year. I have never, ever found the string tree on Fender-style basses to cause the kind of problems you are describing. And even if this was a problem, I don't see how it could possible happen with smooth flatwound strings, unless the string tree in question has some serious issues! I have however experienced poorly fitted nut slots to cause serious problems, LOTS of times.
I'm not saying this didn't happen to you of course, I'm just trying to understand how something like this can have happened on several different basses. With strings that have rougher tapered ends that might (in theory) get stuck on a string tree, like for example Roto 66s, I guess I can see something like this happening once in a blue moon. But I use Rotos on Fender basses a lot myself and have never had any problems with this. I have a feeling that the problem you are describing might in fact be with the nut, especially when you say it has happened on several different basses rather than just one (which could have had an unusual issue with the string tree).
If your solution solved your problem that's great of course, but on a general basis I would say it's a VERY bad idea to raise the string tree and decrease the already poor break angle over the nut on Fender basses. In fact, I usually ADD a string retainer the includes the A-string on my own Fender basses. Raising the string tree on Fender-style basses can cause a poor tone on open D/G-strings and also potential buzzing in the nut slots.
As for the OPs bass, I suspect one of the following causes:
1) If you're using heavier gauge strings than 45-105 the nut slots might be too tight. This can cause the stings to get stuck in the nut slots and break (it can also cause poor tuning stability). Especially if the nut is made frome a poor quality plastic/synthetic material rather than bone.
2) Maybe EB Cobalts are a bit fragile and/or have poor QC. String breakage does seem to be a recurring issue with these strings, based on what I read here on Talkbass (I have very little experience with them myself, having only installed them for customers).
3) Check the tuner posts for any sharp edges. I don't think this is the problem here (especially with this happening on several different basses) but I've seen stranger things...
