So far the three answers are: Yes, No, and Maybe. I feel like I need to chime in here and give a more pragmatic answer. But before I do that, let me give you my credentials so you know who this is coming from.
I design PCB's for a living. Large and complex PCB's. Recently I have designed a custom motherboard based on an Intel CPU as well as several PCI Express boards. One of those PCI Express boards is powered by a supply that is separate from the ATX supply that powers the motherboard. These are embedded systems, which means that I can get away with things that you couldn't in a standard PC.
On to my answer:
Don't do it! If you value your sanity, don't bother trying. It might work, but probably not. The problem is, if it doesn't work then what are you going to do? Do you have the tools and knowledge to debug it? I'm guessing that you do not have a PCI Bus Analyzer or a copy of the PCI Express specification so the answer to that would be no.
Some of the things that might not work correctly:
Some PCIe cards might connect the +12v from the MoBo to the +12v from the PCIe Power connector. This is fine if both come from the same power supply, but not if there is a second supply. In my opinion, this is a bad design but that doesn't matter. You should check the card before trying anything.
There are tight specs for the time from when the power supply comes up until the PCIe card has to respond to PCIe activity from the motherboard. If the two power supplies take different amounts of time to come up then this spec could be violated.
Feeding a PCIe card with 100-200 watts of power is still non-trivial. The supplies themselves are expensive, too. It turns out that the cheapest power supplies available for this are ATX supplies!
There could be some power sequencing issues, where certain PCIe signals come up before the receiver is ready. Although unlikely, this could result in a damaged motherboard and/or PCIe card. If something is damaged, it is likely due to a bad design-- but bad design are out there.
In my opinion, you will be time, headache, and possibly money ahead to replace your current PC and simply get a new one that can handle the PCIe card you want to use. Otherwise, you would likely spend a lot of money on a new PCIe card and a power supply to power it only to end up with something that doesn't actually work. Or worse, you might damage your current PC.
Best Answer
Most of this, as you've probably seen from the comments, isn't actually electronic trickery. They run multiple power districts across the board and drive them from the PCI-e slot and the cables separately. They're all running on a common ground from the same PSU, so it works quite nicely. Usually you'll see the VRAM and peripheral logic running from the PCI-e slot supply, and the GPU itself running from the PSU.
For boards like the GTX260 OC2, which are explicitly designed for overclocking, you'll see cases where two 75W 4-pin PSU connectors go into the same card. When more than one is used, they do use some form of current limiting (can't tell what on my card - it's under the housing) to "combine" the maximum current of the two inputs. This apparently results in a maximum consumption of 225W rather than 150W, which can be useful for people
insanehardcore enough to cool their hardware with liquid nitrogen.Rather surprisingly, a little research indicates that computer power supplies are designed in a way that allows you to directly wire two +12V rails together to get a boost in maximum current, without any damage to the supply. I'm skeptical, but it looks like some nutcases on various forums have done it, and one company even sells a special connector that combines them. If it's a legitimate thing that the ATX spec says they should allow, then I guess certain cards may just join the rails together in a similar way. I wouldn't recommend trying it yourself, though.