Uniqueness of Transient Node Voltages
In the capacitor case, there is a single unique solution and SPICE will find it.

Convergence issues can arise from multiple cases:


model errors -- if the model for the component contains equations that are discontinuous (or are discontinuous in first or higher derivatives), SPICE may fail to converge around those locations
'Stiff' equations -- if you have elements with hugely differing magnitudes -- say 1 \$  p\Omega \$ (pico ohm) and 10000 \$  G\Omega \$ in certain configurations, the floating point representation may not be able to represent all quantities with sufficient accuracy. A related case is strings with very high gain -- e.g. large numbers of inverters in series with DC feedback. In practice this rarely occurs. 
Circuits with more than 1 solution -- e.g. in a perfectly balanced flip-flop there can be 3 solutions -- a) balanced -- each output at VDD/2, b) 1 &amp; 0, c) 0 &amp; 1 outputs. Without hints and starting points, SPICE may come up with the unstable equilibrium balance point. Similar issues occur in (incorrectly designed) bandgap reference circuits. 
(non-physical) Circuits with behavioral models, no delays and 'infinite' gain -- e.g. a Schmitt trigger with input connected to the output. With zero delay in the model, there is no consistent solution.