I would guess, "It would take a very long time--possibly forever--because scientists don't typically become 'worried' by perceived shortfalls in theories, rather they become excited." Often, that excitement finds an outlet through modifications to the theory or to wholesale alternative theories. In the case of Dark Matter, there's no shortage of proposals to overturn Dark Matter. It seems like there's a new one on HN every day. But, "the more's the merrier", I say! If you ask me (nobody did), they're all welcome. The only catch is, they have got to surpass Dark Matter on all of the many independent lines of observational evidence supporting the theory of Dark Matter, yet in my limited experience they almost never do.