The GTX 670 -- at least gigabytes -- can consume 325 watts. The 5870 consumes even more than that (as I note earlier in this thread.
The 4870/4890... well take a look:
Image
As you can see, they consume a lot less.
So yes, these new cards (GTX 670 + 5870) consume significantly more power than the 4870/4890 that you have had "no problems with". I would highly recommend re-examining your setup if you do indeed run either one of these cards off a single 12V lead coming from the drive bay.
I don't know if you'll ever have any issues running this way, but you're definitely pushing the tolerances of your cabling. Even if you have a "reference" GTX 670 with only Two 6-pin plugs, the card is expecting FOUR 12v leads with a total rated capacity of 150 watts. You're potentially shoving that amount of wattage through only 1 lead.
18 gauge wire --JUST THE WIRE (connectors are less)-- is only rated for 7 amps. You're potentially running 12.5 amps through it. [Edit: just found out that in the case of light fixtures in homes, running more than 6 (six) amps through an 18 gauge wire is actually against the law]
Likely the consequences will be voltage variability and heat generation during high-load. It may not lead to fire or melting or anything dramatic, but it's very possible you'll lead to instability.
Now. If you have a Non-reference GTX 670 with an 8 pin plug and a 6 pin plug, that's got a lot more potential. 8 pin plugs are rated for up to 225Watts (18 amps) by themselves. You may be okay if you don't manually overclock, but I wouldn't risk it.
Food for thought.