I have noticed that most those services (twitter, Skype and so on) block login without giving phone number...until few months later they automatically unblock. I simply check out in every week, are they removed block or not.
Once I was stupid and gave phone number one of those kind services, that was bad idea. Account went into some weird state after giving code received from SMS.
So my personal experience: wait, never give phone number.
That's odd. I've never given Twitter a phone no. (Granted my account predates their phone no. fetish.) Still use Twitter daily without any issue. Perhaps it's because I'm still logged in and not trying to re-login?