To build a site that fails in both IE and Firefox has got to indicate a project that's truly off the rails (no pun intended, for other web monkeys out there).

they often demand obscure or obsolete browsers for correct operation

This too is a really bad sign. In a fundamental sense, they're Doing It Wrong.

Most emphatically, the problem is not on your end, TB...but sadly, that knowledge isn't very helpful when faced with a development department that, for whatever reason, can't or won't correct the situation. In an age when mature web development tools and methodologies are plentiful, there's no excuse for directing clients to non-functional web applications.

It's perhaps not the most scientific metric, but when I see a site that's clearly in trouble, it makes me wonder about the stability and reliability of the entire organization. Could be a bias on my part. I'd guess (or at least hope) than many of these organizations are rebuilding their sites, but they've probably also formed Steering Committees, so it could take a while.