You have bad wiring or a bad outlet. Turn off the breaker for that circuit immediately as this type of problem can cause arcing or heating at the site of the break, possibly leading to a fire. And no, this type of failure may not trip a breaker.
Depending on your comfort working with household voltages and wiring, you can start troubleshooting by figuring out where the wire for those outlets start and inspecting the end for corrosion, nicks on the conductor, or breaks in the conductor or insulation. Also try replacing the outlets themselves, including the outlet before the non-working outlets.
A quick anecdote: an outlet in my previous house that had been working fine for years developed this weird behavior where if something was plugged in, nothing else in the room worked. Turns out that the outlet had a manufacturing defect that only became apparent after 4 years of use.