We also put in offers on three other places before we had ours accepted. The first time, they went with a different buyer, then took it off the market. Second time, we offered 5K less than asking price and they said it wasn't enough and later took it off the market. Third time, fabulous apartment, many, many takers, and ours wasn't the highest offer. Oh well.
So, given our negative history, when we found our new place we instantly made a full-price offer. Didn't wait to mull it over - we had seen hundreds of places, so there was no need (although it made my husband very nervous to act so quickly). Didn't try to haggle. We were in a hurry by this point.
What happened: we offered full price, plus we wanted all kitchen appliances. Seller accepted. THEN the agent started pushing his in-house mortgage on us, which had a higher rate than the one we were going with. Threatened that the seller might not go through with it if we're not a "guarantee" (because having an in-house mortgage makes us so much more "guaranteed"). We agreed, on the condition that we get 7.5K off the purchase price. Seller agrees. Mortgage doesn't go through, so we're back to our old mortgage with the better rate AND the discount on price.
Does this count as gazundering? I don't really think so, but it is one of the ways that people can end up with a lower price after an offer's been accepted. I don't think that gazundering is really a normal practice, though. Like the other posters said, it has good potential to backfire.
But. The housing market is in an odd situation at the moment (or at least it was the past six months, while we were buying). Since people are frequently selling for less than they paid, they're often reluctant to let it go at a price that will actually sell. So some houses sit on the market for ages and ages, while reasonably-priced houses are being snatched up the moment they go on the market because they're so rare. We saw this happen over and over and over: we'd find a great place, then *snap*, it'd be gone. It was really, really hard on my husband because he just can't stomach spending that much money, and wanted to think about each individual flat for a week. But we didn't have the luxury to do that, and lost a couple of places because we couldn't even make an offer before it was off the market.
Anyway, to answer your question, yes, Brits and Americans do it differently. Especially in the current market, it's important to look within or under your ideal price range and be prepared to either make a close-to-asking-price offer or be turned down. You can make a few cheeky offers if you want, but in my experience you'll just be told where to stuff it. Also, I doubt they just want you to go away - they're not going to refuse your money because they don't like your previous offer. I've seen the listings for some places stay up after they're sold, while other agents will put SOLD over the ad five minutes after the offer's accepted. Don't worry about it. Just keep looking for the best property for you, and be aggressive when you find it!
Good luck, and happy house hunting!