london bridge is falling down?

No, it's thought that the rhyme is based on the original problems in building a bridge across the Thames. The original London Bridge is now in the USA - the buyer apparently thought he was getting Tower Bridge.

The old London Bridge across the river Thames was built in 1209 and had houses and shops on it.

Various arches of the bridge collapsed over the years, and many of the wood framed buildings on the bridge were burned down in fires.

The present London Bridge was finished in 1972, it replaced a nineteenth century bridge which was removed and rebuilt at Lake Havasu, Arizona, U.S.A.

So you could say it keeps falling down and being rebuilt.

When I was a kid, yes we used to sang it while playing a game called " My fair Lady or boy". We used to make a circle and then all of us hold each others hands, and started to sing "London bridge is falling down", while there's one in the middle with her eyes closed and point the next person to play when the song finished. We used to enjoy this game when we were kids, but I only played it at school with my classmates.