Is it possible for the system of equations of an electric circuit to not have a unique solution?
So long as you consider networks containing only positive-valued linear resistors, ideal voltage sources, and ideal current sources (and you don't put two current sources in series or two voltage sources in parallel) there will always be a single unique solution.
I don't have a proof of this ready to hand, but it is pretty clear that if you follow the (modified) nodal analysis method you will obtain one equation for each node (other than the ground node) that isn't connected to a voltage source, and one KVL equation for each supernode, plus a supernode equation. And that these equations will be linearly independent because each node connects to a different set of branches. (A complementary argument showing a similar result for mesh analysis)
For a thorough proof, see, for example, Chua, Desoer, and Kuh, 1987.
If you consider nonlinear resistors, it is possible to have a circuit with multiple solutions. One way this happens is if the circuit has hysteresis, so that the correct physical solution depends on the history of how the source voltages were applied to get to the situation being analyzed.