Not all rehabs will work out a payment plan for you. However, many will offer various financing options. You might not have a credit card that will allow you to finance the full amount of your rehab stay. However, it's possible that there is a different line of credit that you can apply for. One example of this is a company called MyTreatmentLender.com, and there are many others.
Many financing companies will offer interest free payments for as long as 12 months. This is great because it cuts down on the amount you would need to pay back. Talk with the facility before you commit to going to rehab. Ask them what they can offer to help you as far as financing goes.