I'm on an old Muji 3-speed commuter bicycle which uses an internal Shimano gear hub, and I recently (3 months ago) got my bike chain changed by my LBS.
For about 2 weeks I felt the chain would "catch", the way it feels when you're trying to change gears on a bike with derailleurs, and last night on my way home, the chain broke. Is this because the chain didn't fit well?
3-speed drive trains should naturally have a straight chain line and lots of room for the chain, which is typically wider. So I doubt that the chain is too wide.
I suspect the LBS did not correctly insert the chain pin, when reassembling. If this was the case the pin could be protrude on one side and potentially catch on various drivetrain components. Continually catching will stress the chain. If the protrusion is significant enough to catch, on the opposite side of the protrusion the pin may not be sufficiently inserted into the outer link plate. Eventually the catching could cause the partially inserted pin to pop out of place, breaking the chain, leaving you stranded. Check where the chain broke to see if the pin at the break point is reasonably flush to the outer link plate or protruding by a large amount on the side where the pin is still in place.
This is of course rampant speculation, as I have no pictures to go off of, but is the only logical mechanism that comes to mind. (And it is something I have actually done and can verify can happen!)
When you change the chain you have to change the sprocket also. They wear at an equal rate and a new chain will not fit an old sprocket. The front sprocket is usually ok because the load and wear is spread over a greater area.
Chains usually only break when lateral force is applied to them. To have one break on a single speed drivetrain is fairly rare. I'm gunna give your LBS the benefit of a doubt and assume they installed the chain correctly. The problem with changing just your chain is that it's only one piece of a system that wears as a unit. As your chain wears it actually gets longer, this is a result of the friction from the pins taking metal off of the inside of the bushing in each link, which lengthens the measurable distance between each pin.
As a result, the cog starts to wear on the pull side of each tooth.
An example of an excessively worn cog:
When you put a new chain on a worn cog they won't necessarily mesh. The new chain will have some "wiggle room" to slide around in and possibly even slide laterally enough that one of the sharpened teeth can pick up a link, which on a single speed can put enough force on the one side of the link to pop the pin out.
My question for you would be, how worn was the cog? If the cog is fine, then how loose was your chain? A loose chain can just as easily jump the teeth and break.
