1 If the mortgage is not paid, the lender can foreclose on the home--i.e. the bank can take it.2 Since you are on the mortgage, you are legally responsible for it you can be sued for the money owed and will face credit damage.3 You in turn could sue your ex-husband for any losses or costs you suffered due to his failure to keep up payments and/or refinance, buta if he in good faith tried and was denied refinancing due to, say, unemployment, he might not be liable, since in this event, it would not be his fault--the failure was due to factors beyond his controlb if he is liable but has little or no income or assets, then even if you sue him and win, you might not get much or anything from him a court judgment in your favor does not make money appear where there is none.