Does walking fast prevent eggs hatching?

Based upon my time playing Ingress (created by the same company and also using a lot of the same infrastructure it seems), you can go at least 15 mph (~24km/h) and it SHOULD still track you. (This is a little less than the 30 kph someone in the comments mentioned).

Note that 15 mph is a 4 minute mile (or 2.5 minutes per kilometer). This is a little slower than the record running speed, and still over the average biking speed.

It also doesn't tend to lock you until you've made multiple jumps at faster speeds, and it only does so to stop spoofers.

Therefore I doubt it is your walking speed.

If your game isn't registering your distance, I would assume it is either GPS troubles, the game in general, or server issues. The majority of this is handled by Niantic and will likely improve as the game progresses. Unfortunately, that means there is probably little you can do other than contact Niantic to let them know.

I went for a 5km run and it only tracked a little over a 1.5km. Had the app open the whole time and slowed down or stopped several times to catch pokemon. I dont know if the stopping threw the tracking off or the changing of speeds. I'm hoping it just a bug in the game right now and will be fixed, because I would prefer to go for my daily runs and catch pokemon than have to walk the whole time.

I noticed earlier as I was walking, it stopped showing my character walking. I just closed the app and reopened. Perhaps that could be your issue.