Citing the Wikipedia article:
The orbit's perigee and apogee had been selected to cause reentry due to orbital decay within 10 days (the limit of the life support system function) in the event of retrorocket malfunction. However, the actual orbit differed from the planned and would not have allowed descent until 20 days.
The obvious case where Gagarin would be stuck in orbit is the retro thruster not firing. But from miscellaneous sources ;-) I know one needs very little delta-V to perform a retrograde burn at the apogee that drops perigee enough for aerodynamic forces to grow enough to deorbit the craft. Assuming inability to start the main retro engine, could the nitrogen thrusters of Vostok-1 have provided enough delta-V to reduce orbital decay below the critical 10 days?