The first thing you need to understand is that COBRA is not an insurance plan; it is a set of laws protecting people who have insurance plans. If you should lose your job or become unable to work, COBRA plans require insurance companies and employers to keep the account active for up to 180 days if you elect to use the plan.
If your group health insurance plan is not acceptable for you, you could look into converting the policy to an individual plan. You would have to pay the full premiums, even the portion currently paid by your employer, if any, but you would then have an insurance plan that is tailored more closely to your needs. Bear in mind, too, that COBRA does not apply to individual health insurance plans, so switching to private insurance will not solve the lack of having COBRA protection.
Another thing to consider is that the Affordable Care Act (ACA) will be changing the way COBRA works. Already, employers with as few as 2 employees are being required to make health insurance available, as opposed to the 50 required before the ACA. When the law has gone fully into effect, you may be protected under COBRA even though you work for a smaller-sized company.
Another option you could consider is to set up a Health Savings Account, called an HSA or Medical Savings Account (MSA). This special type of savings account allows you to have a specific amount deducted from each paycheck and deposited into the account. You then use a debit card attached to the account in order to pay your medically-related expenses. When this money is used for accepted medical purposes, it is non-taxable, but if you decide to withdraw some or all of the savings for another use, you will be taxed on the money you withdraw accordingly and may be liable for other fees as well.