Is it possible for the runtime and input size in an algorithm to be inversely related?
Try brute force searching of a key for a cryptographic algorithm.  The more of the key you give it to start with, the less you have to search for.  True that trend stops at the limit of keysize (but that's still monotonic), and there are probably other examples in the field of extensive search where the more input data, the easier it is to prune branches of the potential tree.