Okay, had a think about this and tracked down a couple of dates for you to chew on.
Recently, there have been two major changes to the live site - both of which rely on javascript. The most recent was added on the 23rd of February (1 day before this thread was made). However, it's not currently applicable to anyone outside the senior staff and, based on what I'm hearing, this has been going on for at least 1-2 weeks prior to this date, possibly longer.
As such, I'm willing to discount that cause for the time being.
The second addition is the Google CSE (Custom Search Engine), which sits in a global notice header at the top of the forum. This was added on the 17th January with minor (and inconsequential) style tweaks on the 8th February. You'll notice that when the page loads, the rendering of the site is held up/delayed as the API runs (I'm on a very fast home connection and I can see the delay). This has been an aesthetic irritant for me, but given that the delay is visible on a quick network, I'm wondering if that delay is worsened on a slower network?
Can any of you replicate this password bug on demand? In other words, if we were to disable the Google CSE, would you be able to tell me if it has any effect?