Actually I don't quite get your point. Following what you've written, think of all k random variables following normal distribution. We don't get multimodal distribution even if a1+a2+...+ak=1. Clearly a1X1+a2X2+...+akXk is always normal for a1...ak with at least one of them non-zero.
I guess the very key issue is not with the weights you assigned to the random variables, but their marginals. But how to construct an example? Help me out!