Can you add a 300 watt power supply to a computer that currently runs a 185 watt power supply without damaging anything?
Yes you should be fine. As long as it fits in the computer, you're fine. Replace the old with the new one but keep the old one in case the new one doesn't work! 300 watts is fine for replacing something smaller.
You should run only one power supply at a time. What this means is that you should replace the 185 Watt supply with the 300. Having 2 separate power supplies creates the possibility of having slightly different values for ground, +5V, and +12V DC. This can cause problems with all of your computer components. Don't use both, replace the old one with the better one. <- There are some motherboards that REQUIRE two power supplies, but you probably don't have one. They're server mb's, and the reason you use dual supplies is, if one supply dies the computer uses the live one and notifies the system operator "one supply just cooked, get me a new one pls." This eliminates a point of failure. If you've got one of these mb's, both supplies have to be the same: two 300w, two 750w, whatever.
Really, though, with the amount of juice some of these new video cards pull, I wouldn't even consider installing a 300w power supply in anything except a server, a lot of which are "headless"--they don't have video outs, you control them over your network. Go with at least a 500w supply, and if you might have the need for a dual-head system--two monitors are GREAT, don't let anyone tell you different--go with a 1200w supply.