You have 60 days from the time you get the to elect, and then another 45 days to pay the premium.
The larger issue is that you may be moving out of your company's plan area. If they do not cover people in Ithaca, NY, it does not matter where your former residence is or where you've been, Cornell is where you'll be.
If they have offices or active employees near Ithaca, NY, they will most likely cover you.
COBRA has always suffered from adverse selection since employees can wait for two months before deciding whether to exercise the option, for which they can get back. The rational choice is to remain uninsured until the end of the COBRA eligibility period. If you are sick, then sign up for COBRA. If you are healthy, buy a low-cost policy in the individual market here:http://www.cobrahealth.com/healthquotes.htm
Also make sure you get a Certificate of Creditable Coverage
to negate any pre existing conditions if your COBRA runs out.