It's not impossible but it's going to be very very difficult. Your first issue will be the bankruptcy that you filed in 2010. You are past the seasoning period for your bankruptcy, however the 120 day late payments are almost as bad as a foreclosure in the eyes of many lenders. The credit union where I work will not make a mortgage loan to anyone who has been more than 30 days late on their mortgage within the previous two years.
Your second issue will be your husband's bankruptcy. There are a couple of options when it comes to bankruptcy seasoning. FHA states that you should be two years post Chapter 7. There are hardship exemptions such as if the bankruptcy was caused by the loss of a job, death of a spouse, etc. In some instances you can be granted a new mortgage after only one year since your discharge but the terms and conditions aren't always favorable.
A traditional mortgage lender will probably flat out deny you based on the late payments on your current mortgage. While you've had some harships beyond your control, the previous five to six years of yours and your husband's financial lives do not reflect sound finances. In the period of six years you've each filed bankruptcy and you've been 120 days late with your mortgage at least once. Negative credit after a bankruptcy discharge is a deal breaker for most lenders.
However, it may be possible for you to get a mortgage. If you can find a lender who will not have any overlays or stipulations such as those I've mentioned, FHA does finance down to a 500 credit score with a higher down payment and the down payment you've mentioned would put you in that percentage range. You won't receive an automatic underwriting approval and you will have to document your hardships. The best loan program I know of that could possibly work for you is offered by Quicken Loans. It will allow for a loan one year after bankruptcy discharge, requires a 640 middle FICO of all borrowers, has a five year adjustable rate, and has a maximum LTV of 80%.
Personally I advise you to wait until your finances improve. You are contemplating taking on hundreds of thousands of dollars of debt when you've repeatedly been unable to repay your debts. Again, life's circumstances do happen and I understand. If life's circumstances were to come and bite you again, both you and your husband are ineligible for a Chapter 7 discharge for quite some time. You could file Chapter 13 if something happened, however your husband would be unable to discharge any debt.
