This issue has to do with my low E string on my Godin LG, which just has a tune-o-matic bridge. For whatever reason, the string's core keeps breaking at the saddle, causing it to "unravel", not snap. This happens well before the end of the string's life cycle, typically after about a week of playing.

What in the saddle would cause this problem and how can I fix it?
There could be sharp edges or irregularities on the surface of the saddle. If so, carefully file them down.

Or maybe you're just sweating acid or something.
Most likely a sharp spot in the bridge. Get a small file and carefully smooth it out.
Emery paper.
Gilchrist custom
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My band
I'm going to go with Cath's suggestion, since I don't want to buy specialty files from Stew Mac.