I have noticed strange behaviour with the Dissolve tool so tried a test on three line segments, snapped to the same central point. The line segments each consist of exactly two vertices i.e. the end vertices. There are no vertices in between.
When I ran the dissolve tool on this (multipart deselected, for testing purposes), I got a new feature class consisting of five records: two of the original line segments had been split in two! I was expecting to get a new feature class with a single record, as all the lines shared a vertex. At the very least, I didn't expect to get more records in the output feature class than were in the input feature class.
Anybody have any ideas as to why it might be behaving this way?