Keyboards can only register so many keys at a time, and the cap changes depending on the keys used. For me, using the arrow keys and S to jump / D to shoot fixes it.
Needs to be mentioned that it's also different for many keyboards, since they are not all wired the same.
I tried it and it seemed to work fine on both my laptop and stationary computer (although I find the use of WASD for a platform game, not using the mouse, awkward).
The, probably, safest way to be sure your keys dont cancel each other out is to use arrow keys + Z/X or arrow keys + S/D, or something to that effect. I usually consider the arrow keys in combination with a, s, d and/or z, x, c, as safe.