The one-dimensional complex Ginzburg-Landau equation (CGLE) with a destabilizing cubic nonlinearity and no saturating higher-order terms has stable bounded solutions. We consider a simple pedagogical model exhibiting qualitatively the mechanism which may suppress the divergence of the solutions. Then we investigate the functional form of the blow-up (collapse) solutions immediately before the divergence. From this analysis we find analytic boundaries for the existence of collapse solutions in the parameter space of the CGLE. A comparison with numerical simulations demonstrates that for parameters without collapse solutions the solutions of the CGLE remain bounded for all times. Finally we discuss the implications of our results for the solutions of the CGLE when saturating higher-order terms are included.