If you look at it from the "did he do what he was swore to do" perspective, the answer would have to be yes, he failed. However, I think Tolkien wanted us to look beyond that strict definition of failure.

If you undertake a formidable task, remain focused on the task, overcome numerous obstacles to completing the task and in the end are over-whelmed by forces beyond your control, have you failed?

For the answer, we have look within ourselves.
Would you or I believe that we had failed? If we so believed, would we let the memory of that failure eat at us, perhaps as Frodo may have done?