I will be leaving my current employer in one week. I will be starting with my new company immediately. I will lose my health care benefits the day I quit my current employer, and will not pick up my new coverage for 1 month.

My question is what are the rules on COBRA retroactive? For instance, if I elect not to carry COBRA and God-forbid something terrible happens, like a car accident, can I go back and pay my premium, retroactive, and be covered for the car accident?

I'm guessing that I am stretching it here, but am trying to avoid a $950 extortion fee from my employer.

No, once you refuse it, it is refused. Cobra basically is the full price of the insurnce ( the part the company pays and your part)

There are many companies that sell one to three month policies. They do not cover pre-existing but will do all the rest.

If you are in TN I would be glad to give you a list of many good ones here.

There are also accident plans, that pay only in the case of a accident,

I'm afraid Iwillhelp and Chcuk are totally wrong here. You have 60 days to decide whether to pay the COBRA premium. As long as you pay it within those 60 days coverage is extended.

The following is a quote from the law:

The election period is the time frame during which each qualified beneficiary may choose whether to continue health care coverage under an employer's group health plan. Qualified beneficiaries have a 60-day period to elect whether to continue coverage. This period is measured from the later of the coverage loss date or the date the notice to elect COBRA coverage is sent. COBRA coverage is retroactive if elected and paid for by the qualified beneficiary.

See here for more:
Text of What is COBRA (http://www.cobrainsurance.com/COBRA_Legal_Text.htm)