I fell and broke my hip in a chain store, do you think I can get medical expenses covered

As Ms. Anderson suggests, if there was fault on the part of the store, they may be liable in whole or in part. Simply lifting a heavy item and having a cart move does not necessarily imply fault on the part of the store, unless there was a defect in the cart. However, if after going over the facts with an attorney there does seem to be a case, then you could file suit and medical expenses could be part of your claim. However, this is not like an automobile accident where medical bills are covered under "no fault" insurance. Unless you can prove fault on the part of the grocery store, it is to be expected that the store would not readily cover medical bills without going to court.