Depends on the age of the PSU, and how the 12V rails are configured. A new 400W PSU would probably be fine, but capacitors age and you might find that what was once a 400W PSU is now only a 310W PSU.
In-Win PSU's are effectively no-name generic, although the 80+ standard is a sign that it's not ridiculously low-quality.
I recently had a good-quality 380W Antec fail to provide enough juice for a stable boot using an HD7850. Lesser graphics card was fine, 7850 and a new PSU was fine. It was four years old and I'm guessing the capacitor ageing meant it couldn't feed a modern 130W graphics card anymore.
Some people ask me why I have always enclosed my signature in spoiler tags; There is a good reason for that, but I can't elaborate without giving away the plot twist.