As the problem's not universal, I guess the answer's going to be that the site is broken, but I feel I've had this problem before (different site), and fixed it - but I can't remember how!
A site I've been using for years (but not this one) has a tick-box, to say: "Remember me on this computer". This used to work, but no longer does.
It's a pain, as the usernames and passwords are assigned, not user-selected, so I can never remember either of them, and have to go back to the welcome e-mail every time (lucky I've kept it).
I thought it might be a problem with corrupt cookies (Chrome), so cleared all the cookies and logged in again, once again ticking "remember me" (I'm not being offered the Chrome option to remember password).
But next time I visit, I'm logged out again, and the form's blank. Technically, I think it's correctly logging me out after two hours (login accesses potentially sensitive data, so should have an enforced timeout, for safety). So it's not that I should be staying logged in, but that the form should stay pre-filled, which is not quite the same.