You should always worry about running out of ammo. Always. That's why the second line of that street proverb is "Conserve ammo." Even if you have five spare clips for your Predator, and five more for your Assault rifle, you still should worry about running out of ammo. Why? Because sometimes things get out of control. And sometimes you need to do things like lay down suppressive fire, which eats ammo like mad. Sure, you shouldn't go through more than a clip or two in a single fight, but who says there's going to be only one fight? Moreover, once the gunfire starts, it tends to build from there.
I'm not saying that guns are useless, but in stealth and close-quarters situations, melee is a damn good thing to have. It is even better when you're in the unlikely situation of "Oh crap, someone decided to go Romero on us!" Be prepared for situations like that, and you won't find yourself running empty, and wishing you had some melee skills to fall back on.