According to Kant, one can do what is right and that action still may not have "moral worth." True or false?

The statement is indeed true. This is because, for Kant, what matters is not just the intrinsic good of the act concerned but the motivation behind it. It is possible to carry out nothing but good acts, yet each and every one of them will still lack moral worth if they're performed on the basis of ulterior motives.
Kant uses the example of the fair-dealing merchant. The merchant may have gained a stellar reputation for dealing fairly with his customers—such as not over-charging them, for example—but that doesn't necessarily mean that his actions have moral worth. For all we know, the merchant may have only been acting fairly for purely selfish reasons. Perhaps he's calculated that it's in his best interests to be fair to his customers: that he'll lose business if he tries to cheat them.
Unless the merchant acts fairly out of duty, then it cannot be said that his actions display moral worth. And the same moral standard applies to each and every one of us.