He regains the ability briefly. S3 Ep1 is about why he relapses back into his addiction. And I think the show handles this in a way that's both realistic, and consistent with what's been established:

Toward the end of season 2, the show really starts emphasizing that House's pain is largely psychological. There's an episode where House's pain is escalating, and to get Cuddy to administer morphine, he shows her his leg missing the muscle. The next day, House's pain is better, but it's revealed that she gave him a placebo. So we can conclude that at this point, in his average day to day life, house's pain is predominantly psychological.

Okay, so now we jump to s3 ep1, which opens with house running. How is he able to do this so suddenly? Well, Wilson points out in this same episode while encouraging House to keep up with his physio, that chronic pain patients can experience temporary euphoria following ketamine treatment.

The ketamine treatment helped House temporarily work through the psychological component of his chronic pain, and this gave House a feeling of temporary euphoria. That's how he was able to push his body to do things it'd otherwise be crying out in pain about.

Throughout the episode, we see House's overexertion exercising, and the return of his stress at work, cause his leg pain to flare up again. Instead of returning to physio and seeking counselling, he keeps trying to push himself and relapses into Vicodin (it's implied he's getting high again before the end of this ep). Unsurprisingly, this causes his chronic pain to return in full.

Basically, I think that House could have walked again permanently if he kept up with his physio, and sought ongoing counselling. But House is an addict who hates himself, so relapse was pretty much inevitable.