I have a DNA match on Ancestry.com that shows a possible 1st or 2nd cousin match at the level of 684 cM's (just barely inside the 680 - 1150 range) and across 29 strands of DNA for 1st cousin. There has been a suggestion that my father is actually the grandfather of the match person.
At that level is that relationship even possible?
Please refer to the article The Limits of Predicting Relationships Using DNA. The 684cM of shared DNA precisely complies with the Group C or with the Group D with a little skew. So it is very probable that your guess is correct. But to prove it, of course, you need more tests, because single test can't distinguish who is uncle and niece/nephew, for example. It doesn't give clues about age or generation of person. Just only the distance between two persons to most common ancestor.
If you are interested please write more information: what is the sex of tested persons (I believe that you are man) and from whom you could get additional tests. Also it is very interesting if there are Y-DNA haplogroup and mtDNA haplogroup data in your and this person's test. It may give some additional ideas about relationship.
To follow up on the earlier answer -- one of the problems with using AncestryDNA is that there is no chromosome browser. It's difficult to determine what the relationship is from the raw numbers alone -- you have to use that information along with other pieces of data in order to solve the puzzle.
Compare your situation with the scenario recently published in Judy G. Russell's post, "DNA doesn't lie!" on her blog The Legal Genealogist (posted Oct 1, 2017).
Referring to Blaine T. Bettinger's keynote address at the Professional Management Conference of the Association of Professional Genealogists, she says:
Blaine ... showed an example of a case where each and every one of us, looking at the result, would have concluded that the two people tested were parent and child. The one-to-one comparison of DNA at GedMatch showed 3586.7 cM of DNA in common and a 1.0 generation difference between the two.
The two people were aunt and a niece. The missing information: the aunt was the twin of the niece's mom -- making it appear that the two tests were from parent and child.
She also says:
[Blaine] reminded us that DNA alone can never be enough to prove a genealogical relationship. There’s got to be at least one more piece of information to be able to properly interpret the DNA evidence you get.
In a comment to the earlier answer, you said:
I am in the process of getting retested. Would you recommend they do so also, and if so using the same provider (Ancestry) or a different one?
To make an informed decision, you could try learning more about the individual tests given by the different companies. Some resources that might help are:
Think of what tests exist and how they might solve your problem. Y-DNA and mtDNA tests allow males to trace the top and bottom lines of the standard sideways pedigree chart -- see the illustrations accompanying the article on Mitochondrial DNA tests at the ISOGG Wiki. Both women and men can take mtDNA tests, but what you'll learn is only the line from mother to child (men receive mtDNA from their mothers but don't pass it on). Autosomal tests like AncestryDNA give you information that could come from anywhere in the pedigree.
mtDNA tests might add more information, but before you spend the money, make sure you understand what kind of information the test will show you. The article MtDNA testing comparison chart on the ISOGG wiki also includes links to their articles Before You Buy and Choosing a DNA testing company.
Another way to get an idea of whether a particular relationship is reasonable or not is to examine the case studies of other genealogists using DNA as part of their analysis, such as Dr. Thomas W. Jones's presentation "Systematically Using Autosomal DNA Test Results to Help Break Through Genealogical Brick Walls" which was presented as part of the BCG's Quality Day of Education on 6 October 2017. The webinar was hosted by Legacy Family Tree Webinars and the case study will be published soon in an upcoming issue of the National Genealogical Society Quarterly (NGSQ).
In his presentation, Dr. Jones talks about how the amount of DNA people share can vary from the expected percentages and how the apparent distance from the Most Recent Common Ancestor (MRCA) can vary, because of the way DNA is passed down.
