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I'm American and it drives me crazy to hear this, but it is regretfully not terribly uncommon.

Heh. Muphry's law (https://en.wikipedia.org/wiki/Muphry's_law) demands that you use 'regretfully' instead of 'regrettably' (http://www.oed.com/view/Entry/275034) when chastising someone's language.

It's a regional thing, more common some places than others. Australian English speaker, lived in Chicago for 10 years, it's very common there.

My best guess for the origin was as an opposite to "on purpose".

It's pretty standard usage in the Chicago area of the Midwest.

It's associated with being uneducated and/or working class, at least in Canada. Saying "on accident" here would make you seem very blue-collar, or like a hick.

Do you British English speakers use the phrase "on purpose" ?

It's not proper American English either but it's becoming increasingly common.

Not becoming, I heard and used it as a kid in the sixties.

Amazingly, there is even a paper on the subject:

http://www.inst.at/trans/16Nr/01_4/barratt16.htm