http://www.accio-quote.org/articles/1999/0399-barnesandnoble.html

Q: Will there be, or have there been, any “late blooming” students in the school who come into their magic potential as adults, rather than as children? By the way, I loved meeting you, and hearing you speak, when you came to Anderson’s in Naperville. I can hardly wait until you tour again.

A: Ahhh! I loved the event at Anderson’s. It was one of my favorites. That is completely true. No, is the answer. In my books, magic almost always shows itself in a person before age 11; however, there is a character who does manage in desperate circumstances to do magic quite late in life, but that is very rare in the world I am writing about.

Related, on JKR’s site, the following appears:

Section: Extra Stuff
SQUIBS
I have been asked all sorts of questions about Squibs since I first introduced the concept in ‘Chamber of Secrets’. A Squib is almost the opposite of a Muggle-born wizard: he or she is a non-magical person born to at least one magical parent. Squibs are rare; magic is a dominant and resilient gene.

Squibs would not be able to attend Hogwarts as students. They are often doomed to a rather sad kind of half-life (yes, you should be feeling sorry for Filch), as their parentage often means that they will be exposed to, if not immersed in, the wizarding community, but can never truly join it. Sometimes they find a way to fit in; Filch has carved himself a niche at Hogwarts and Arabella Figg operates as Dumbledore’s liaison between the magical and Muggle worlds. Neither of these characters can perform magic (Filch’s Kwikspell course never worked), but they still function within the wizarding world because they have access to certain magical objects and creatures that can help them (Arabella Figg does a roaring trade in cross-bred cats and Kneazles, and if you don‘t know what a Kneazle is yet, shame on you). Incidentally, Arabella Figg never saw the Dementors that attacked Harry and Dudley, but she had enough magical knowledge to identify correctly the sensations they created in the alleyway.

Aunt Petunia is another common guess and Jo has this to say:

Section: Rumours
Aunt Petunia will start exhibiting magical tendencies
No, she won’t. Aunt Petunia has never performed magic, nor will she ever be able to do so.

and

Is Aunt Petunia a Squib?

Good question. No, she is not, but—[Laughter]. No, she is not a Squib. She is a Muggle, but—[Laughter]. You will have to read the other books. You might have got the impression that there is a little bit more to Aunt Petunia than meets the eye, and you will find out what it is. She is not a squib, although that is a very good guess. Oh, I am giving a lot away here. I am being shockingly indiscreet.

Other characters Dumbledore could have misjudged are Argus Filch and Mundungus Fletcher, but I don’t think that could be a pivotal case either. (Although I’m sure Mundungus will create some problems for Harry since he has probably stolen the locket Horcrux from Grimmauld Place.)

What I first thought when I heard that Dumbledore was giving J.K. Rowling a hard time was that 1) Harry was going through emotional turmoil with the portrait, or 2) J.K.R. is struggling with Dumbledore either within a memory, a time-turned era, or another form that isn’t a ghost or a portrait.

I can’t pin down any concrete thought on this subject. On a related note, however, I believe that Fawkes will play a more significant role in Deathly Hallows. Perhaps there is a lingering connection between Fawkes and Dumbledore? I could be wrong, but for some reason I recall reading that Harry will “have a new pet” in an interview somewhere.