Can dissolved salts (or any soluble chemicals, for that matter) 'bump' each other out of solubility?
Yes, this effect is termed salting out for non-ionic compounds and the common ion effect for ionic ones.

Note that in specific cases, the converse may hold true. For example, "the solubility of silver chloride in water is reduced if a solution of sodium chloride is added to a suspension of silver chloride in water (op cit)." However, adding more chloride re-dissolves the precipitate due to the formation of complex ions:

$\ce{AgCl(s) + Cl- (aq) -&gt; AgCl2- (aq)}$