When a witness won't answer your question directly, you should keep on track and ask again, like this:
After a couple rounds of repeating the question, you'll usually get a proper answer. Sometimes, though, you'll have to increase the pressure with an "unresponsive" objection.
In my experience, witnesses often become unnerved when you "move to strike" their answer and immediately become more cooperative. Even if they feel comfortable about the deposition process in general (doctors, for example), your motion to strike leaves them standing on uncertain ground, possibly feeling as if they risk suffering some personal sanction.
Although the "nonresponsive" objection can also be useful at trial, keep in mind that the objection can only be used by the questioning lawyer. As explained by Joseph M. McLaughlin in an article in The Litigation Manual: Trial--
Nonresponsiveness is a problem between the questioner and the witness. It is none of the adversary's business. In other words, the only personal who can move to strike a nonresponsive answer is the person who put the question.
See "Objectionable Objections," Joseph M. McLaughlin, The Litigation Manual: Trial (ABA 1999). See also McCormick on Evidence, p. 52 at 127, n.6 (3d ed. 1984); People v. Sweeney, 46 Ill. App. 3d 858, 361 N.E.2d 344 (1977) ("The statement could only be stricken on a request from the questioning counsel, the only one who has standing to object to the nonresponsive nature of the witness' remark").