COBRA continues your present insurance just as it is now. So, deductibles and out-of-pocket spends continue through 2016. No reset with COBRA. Everything is reset on Jan 1 2017 though, just as if it were regular insurance. Any progress towards meeting plan deductibles in 2017 would not count for anything once COBRA finished. If you choose 12mo COBRA that would take you to 12mo after your termination and deep into 2017. You have the option of running COBRA (at your expense) until 18mo after termination, maybe close to the end of 2017. That self-pay COBRA may be worthwhile if you reach the 'free' after out-of-pocket realm. So, as you see, COBRA has certain penalties with a high-deductible plan vs. COBRA with a traditional subscription/co-pay plan. You do need to be aware of cumulative tax-year achievements which are worthless once COBRA ends or you find other insurance.
It is a gap in the COBRA legislation which was intended to protect employees. It really needs updating to compensate employees appropriately with tax-year thresholds which are important with HDHPs, vs. the simple co-pay model of older plans.
What does change is the (limited use) FSA you may have. You would need to google because it gets complicated, but monies paid into the FSA before termination may only be claimed for expenses with a date-of-service before termination date. You can continue FSA into COBRA, but you would need to pay in with after-tax dollars in order to extand the FSA period and be able to claim against the funds paid in before termination. It gets painful, especially when the Hewitt folks explaining it don't know the rules. I ended up losing all my FSA money for the want of not being told that I could only use it if I (for example) contributed $1/mo until the end of the year. Not knowing about that $7 cost me $500. Argh.