How can I power my 6-pin PCI Express card without using an ATX power supply?
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.