Cell membranes are inelastic. If a cell is flacid (not full) the membrane will wrinkle like your skin if you are in the bath for a long time. When the cell is full, or turgid, the cell membrane is stretched tight.

It is possible for cell membranes to burst. One such scenario is when it is infected with a virus. The virus uses the cell mechanics to replicate until there is no space left and then the cell bursts. Another scenario, although less likely in the body, is if the cell is in a highly saline solution.