Lenders don't care why you made late payments... merely THAT you've made late payments.
A history of late payments (and that's what your credit score represents) is an indication as to how you treat your obligations to those you've borrowed from.
That being said... while your scores are low, they're not horribly low. Some lender should be willing to lend to you, assuming that your scores are the only issue. There are other issues that lenders look at. Amount of open credit, credit-to-debt-ratio, age of credit history.
Shop around... if the lender tells you NO... don't make excuses... ask them "why", and listen carefully.