If you have any kind of tweak or injury, no matter how minor, plenty of guys won't run it. If they suspect they can't get a representative time, they are advised not to do it. Not running the 40 is better than running a slow 40. LSU has had a bunch of guys decline for that reason over the years, none of whom were slow in college or in the NFL. They may also decline some drills at the combine if they already have recorded times at their school's Pro Day. Many of our guys have clocked times they're happy with at our Pro Day that the NFL teams accept and choose not to repeat the drills at the combine. It's not a thing that hurts them.
Rumors, on the other hand, can cause guys to slip appreciably. Some team hoping to get a steal by causing a player to drop to a lower slot.