Is It True That Lip Bumps Can Dissolve Naturally?

Most lumps will decrease over time, but some do remain persistent.  Lip filler bumps is fairly common after filler.  The bumps can occur early on and many settle down.  Although massage after filler is considered a mainstay, beware that massage can drive the filler into another location if done too early.  
Reasons for bumps after filler:
Technique- Filler may have been used too superficially or wrong type of filler
Mouth movements- In some patients their movements can promote the accumulation of bumps and an alternative technique can be advocated.
Treating lip filler bumps:
Observation +/- massage-  Should be done for almost all lip filler bumps for 2 weeks.  For our patients we usually advocated massage after 48-72 hours to let the filler settle into place.
Reversal- A reversal agent can be used such as Vitrase to dissolve the filler if there is a large obvious persistent bump
Additional filler- An expert filler injector can used improved technique to mask the lip bumps.
The application of lip filler is very technical and should be done by an expert injector.

I'm sorry to hear about your experience. It is true that lumps caused by hyaluronic acid based fillers like Juvederm and Restylane will dissolve on their own. It generally takes a few months to do so, however, so please be patient. 

Lip bumps after filler are most likely filler material itself., They do eventually dissolve but it can take 6 months or longer.