It would just depend on what you mean by "similar meaning". For example, "right" (as in human right) and "rite" (as in rite of passage) have always seemed pretty similar to me in some sense. Close enough? They're certainly not equivalent, but few words are, regardless of origin. Something else to consider would be various folk etymologies for words that seem related but are not. For example, "duck" as a verb and noun is thought to be related to the action that the animal does, but there's no historical connection. Those aren't quite close enough, but maybe you'd find something. There is also more discussion of false cognates across languages-- consider Japanese namae vs English name (same meaning). And other disputed cases like arigatō (thank you) in Japanese potentially being from Portuguese obrigado. Probably something like this could exist in a language, but it would be rare as freknu says. The other problem is that words don't have singular origins. They are the result of complex generations of usage, where the very fact that they sound alike might start to make them blend. It would be interesting to find cases where unrelated words have actually blended together.
The only other thing that comes to mind (sort of an inverse of what you're talking about) is when two cognate words are borrowed into a language. English bank and bench are from the same source, borrowed very differently. Originally it referred to a tilted surface. Similarly things like shirt and skirt. They mean different things but have the same ultimate origin with different paths of borrowing/inheritance.
