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I didn't think so, but maybe I went slightly lighter. Ashamed to say the old strings had been on since I bought the guitar new.
Sparky-
ok, if you haven't put heavier strings on I think you may have put too much tension in to the strings causing the bridge to be pulled up. so the issue isn't with the springs holding back the bridge, its the opposite. floating bridges are a bit of a balancing act and can be a pain to set up.
in that case I would slacken the strings right off, get the e strings roughly in tune then tune up the other 4 but marginally increase the tension on each string, working from the outside going in, much like cutlery. once they're at the right tension try retuning it but keeping in mind you might throw the other strings out of tune, so don't go too far on each one.
other way is to block off the tremolo with a bit of wood and tune it up, then take the block out and retune it. that would be the easy way to do it at least.
I've never owned a strat but used to have Floyd rose trems to set up and this was usually what I did. once it's set up it's usually pretty stable
Dogtemple
Have you changed string gauge? That can cause the bridge to lift like that. Also tuning can do that. If you find your continually tightening the strings to get it in tune