I recently fell on my arm, and it has hurt for the past couple days. at what point should i suspect a broken or chipped bone?

Not without x rays: It may not make any difference in treatment of your codtion but without x rays it is not possible to say about fracture.

Time to get checked.: If it is still hurting after a couple days -- and especially if it is getting worse -- it is time to get it checked. It may still be a bad sprain but there are many types of "small" fractures (or breaks) that need attention or they may not heal properly. You should start with your primary doctor or with an orthopedist if you have seen one before. Good luck with your recovery!