The DNA tests are usually quite accurate, and it is likely that one of your genetic grandparents indeed has the differing ancestry.
If you are in the situation that you believe you know who all your grandparents are, and that the ethnic connection does not seem possible, then this is likely what is best known as (in genealogical DNA analysis) as an NPE event, i.e. Not the Parent Expected.
There are several different ways this can happen: adoption, infidelity, or a crime of a sexual nature. Often, especially for our grandparents' generation, these events were covered up and children were raised as if they were the true children of both parents. In some cases, the other parent may not even know they were not the true parent.
Yes you can find out which grandparent was not the true grandparent. Before you do so, ensure that there is nobody who would be hurt by this knowledge, and that your family approves that you investigate. Once you open up a can of worms, it can't be closed again.
If you are satisfied that finding the answer will not hurt or offend anybody, this is what you can do:
First, check your own results:
Find out through the DNA testing company you worked with, what other testers match some of your DNA and seem to be related to you. If you know they are related, or if you can see by their research that they connect to one of your grandparents, then you can be fairly certain that their connection is through that grandparent.
If you have enough matches from that DNA service, you will likely find that 3 of your grandparents have matches and one doesn't. The other people are likely related to your true genetic grandparent, and they'll be people you don't recognize as relatives. That "blood" grandparent will likely be the one who is supplying the different ethnicity. The grandparent without matches was likely the one who raised your parent, and may or may not have known that they were not the genetic parent.
Second (if necessary), go the extra step:
If you don't have enough relative matches from your own DNA test to determine the grandparent, and you still are confident enough that disturbing the family history will not cause animosity, then you can do this: Find one of your cousins that you know/think is related to each of your 4 grandparents on only that grandparent's side who is willing to do a DNA test.
When you get the test results, the relative who does not have a 1/4 match to your DNA results will indicate which grandparent is not your genetic grandparent. Finding common relatives between that cousin and yourself will help you start to build a family tree and ultimately the identification of the unknown grandparent.
You can do this before or after. And you don't have to tell anyone either: Find pictures of your 4 grandparents. Find pictures of their children or grandchildren. See if you can identify the one grandparent that has the fewest physical features in common with the children and grandchildren. If one grandparent doesn't seem to have passed on their appearance, they likely may not have been the genetic grandparent.