Can you get a nosebleed from eating spicy foods?

Yes, but generally this is only a problem for people with a pre-existing susceptibility to epistaxis (the medical term for nosebleeds). Spicy foods, dry air, and high blood pressure caused, for example, by vigorous exercise can all cause blood vessels to rupture and cause bleeding of the nose. Factors for susceptibility can be many, including congenital defects, previous trauma (such as a recent episode of bleeding), vitamin deficiencies, or use of some drugs.

For further info, see http://www.hmc.psu.edu/healthinfo/no/nosebleed.htm and http://www.webmd.com/a-to-z-guides/occular-hypertension.