How many weeks? Generally you have 60 days to elect coverage for COBRA after the triggering event. If you will be covered under another plan within this 60 days then don't elect. Let's look at it this way for example:
If you are coming up on the end of you waiting period for your new plan to take effect and 7 days before that date your kid breaks his ankle in such a way that surgery needs to be performed (screws and plates). What you would do is then go back to your ex-employer (or whoever administrates your old plan) and elect the COBRA (as long as it is within 60 days or the triggering event). They then would notify the Insurance Company and you would be billed the amount due the insurance company from the date of the triggering event, thus, no lapse in coverage. The plan would then pay for the surgery. Remember that the COBRA costs are what the employer pays for the plan plus an Administration fee (2%), so you would pay 102% of the cost of the insurance (companies usually pick up most of the cost of the plan, not so with COBRA). It all depends on how long the gap is. You may not even have to get the COBRA, as in, if nothing happens to warrant a claim during the waiting period. Look at it as retro-active in this case, no claim means no election means no premium payment.
There are individual Gap policies (normally disaster type policies) available form most companies (Celtic, Megahealth, BCBS etc.) but they are typically last resort and not very benefit rich.
Sorry for the ramblings.... and to completely confuse everyone.