This is another "maybe this will help someone" kind of post. Long story short--if you stop getting login verification text messages from twitter, try texting GO to 40404.
At some point I enabled the login verification feature for my twitter account (@carl_welch). So whenever I want to log in to twitter.com on a PC or laptop, I type my username and password, and then twitter sends a 6-digit numeric code to my cell phone via SMS. Then I type that code into the form on twitter.com, and I'm logged in. I have my browsers set to delete cookies when I close the browser, so I end up doing this somewhat frequently.
(It works a little differently logging in to the twitter app on my phone. I have to log in to twitter.com on a laptop or PC and generate a one-time password that I use for logging in on the app. I generally only have to do this once, because I don't log out of the app on my phone.)
I got a new phone a couple of weeks ago: same number but a different provider. And I immediately stopped getting the login verification messages from twitter. So I couldn't log in. I'd managed to get logged in on the phone's app before restarting the browsers where I was logged in to twitter, so I could still use twitter while this was going on. But before I realized that I had a problem, I'd restarted my browsers, and I couldn't log in (on a PC or laptop).
I submitted a help request to twitter explaining the problem, speculating that it might be because I changed provider, and asking them to disable login verifications for my account. I'm quite disappointed that--other than automated responses--I never heard back from them at all. Lame.
So I kept poking around on their help pages, and I stumbled upon a page talking about how to tweet via SMS. Seems like you start that process by adding a phone number to your account (which I'd already done) and then texting GO to 40404. For lack of a better idea I tried doing that on my new phone. And login verifications started working again immediately.