Can you assume someone else's debt

What do you mean exactly? Like put the debt in your name to help someone else out or do you mean can it get passed to you without you wanting it to happen?

I mean take on a family members debt in your own name not theirs because of their in ability to pay.

I doubt if a creditor would release any debtor from their obligation. But what you CAN do, is ask to be added to the debt as a guarantor. You can then bring the account up to date and maintain payments. This is even better since this will help rebuild the relative's credit.