Depending on the type of loan you're thinking of, the amount of down-payment you can make, your other assets, you may or may not have a difficult time. Also, the definitions of "high DTI" and "good credit" mean something different to mortgage professionals from consumers.
To answer properly we would have to know all these things.
High DTI would be above 43%, but some lenders will still approve you up to about 48% with other factors, like down-payment and other assets. If your new house payment will cause payment shock, or raise your DTI above this, then your chances are slim.
Good credit would be above 680, but some lenders using FHA will approve you down to about 620.