Yes and no. You can't react two insolubles - once it has precipitated it will stop reacting. However is it possible to react two soluble compounds to get two precipitates? That's a good question, you can skip to the bottom for one such equation, but the full answer really gets into the guts of...
OK so there is a little bit going on here. NaCl(aq) + CaCO3(s) won't react at all.
The equation you are talking about is probably CaCl2(aq) + Na2CO3(aq)--->CaCO3(s)+2 NaCl(aq)
What is happening here is that the reactants are all aqueous, so they are all ions floating freely in solution, ie...