A pendulum hung from a stiff arm will always swing in a circle as it is constrained to the circle by the stiff arm. When the pendulum is swung such that the arm passes through horizontal (and is not just spinning rapidly in one direction) the arm will be under compression rather than tension. To see this, imagine a pendulum balanced with the bob above the fulcrum, the force on the bob must be upward meaning the arm must be pushing, and thus in compression.
Now lets replace the arm with a string. Strings deform when the tension is less than zero. Thus once the pendulum would require compression on the arm, the string will deform and the bob will deviate from the circular path.
This is similar to what happens when one tries to swing higher and higher on a playground swing. Eventually the chains start losing tension and then snap tight when the swing falls back down. This process consumes energy, so pumping one's legs with more energy will just increase the magnitude of this energy loss mechanism without much further gain in height.