Pressing 'Up' and 'Left' together gives a beep and does not move my character, while all other direction combination keys work.

Questions & Answers
1

Up, Down, Left, Right, Down + Left, Down + Right, Right + Up all work, but Up + Left doesn’t. Any reason for it to work in 7 directions out of 8 ?
EDIT: Forgot to add that I’m also pressing ‘Space’(for spawning a blast radius, which is a different script) along with all of the above combinations, so there are essentially 3 keys pressed at any time, namely Up + Right + Space, etc.

2

This is a hardware issue, not a Unity issue. This article explains the technical constraints behind cheaper keyboards:

http://en.wikipedia.org/wiki/Keyboard_%28computing%29#Control_processor

I’ve got a CMStorm keyboard and do not have this problem. Unfortunately, the only fix is to buy a better keyboard.