A lot of it depends on the PSU Make/Brand you will want at least 28 Amps on the 12 Volt rail to run the 1950 Pro and the GT is going to be a little less but still i doubt it will run on that PSU.
Without upgrading the PSU you would be ok with a 7600GT as they are very good on power.
It is totally possible that if you put the card in an underpowered system that you COULD blow the whole system or the card may just throttle down to run on available power or just not work at all. Either way its a Very bad idea to just suck it and see, putting a card in a system you are fairly sure is underpowered is just a recipe for disaster.
Best bet would be to get a good 450/500 Watt PSU with a 28+ Amp rating on the 12 Volt rail Antec are good in my experiance and then get the 1950 Pro to go with it.
If you want to check your amps its written on the side of the PSU (just in case you wasnt sure).