Each time I load the page, I get different results so here's a summary what's generally happening when debugging the page:
Firefox
Works fine, every time
Chrome
Error #2035 before page fully loads
GET request fails on assets/locale/en_US.swf
Refresh the page a few times, randomly works fine
Clear cache, back to error #2035
IE
Error #2036 before page fully loads
GET request on fails on assets/locale/en_US.swf
Refresh the page, log in box appears, but WidgetManager popup tells me Error #2036
Debugger says widgets/WidgetContainer/WidgetContainer.swf failed to load
Log in anyway, try to launch a widget, that particular widget's.swf file fails to load
Refresh the page a few times, site randomly works fine
Clear cache, back to error #2035
So it seems certain files are failing to load, which is expected given the #2035 and #2036 errors, but which files fail to load and when and on which browser seems completely random.