I'm making this an answer to make it more visible. Wadim Zudilin has been running a computer program of mine on a fast computer. Today we found a string of length 11. The form is $ 3 x^2 + x y + 26 y^2 $ of discriminant $\Delta = -311.$ The numbers represented run from 897105813710 to 897105813720. Note that this is the longest possible string for this discriminant, as the first quadratic nonresidue $\pmod {311}$ is $11.$ The first number in the string is $\equiv 0 \pmod {311},$ indeed 897105813710 = 2 * 5 * 311 * 288458461. So at this point I conjecture that there is NO general upper bound on the number of consecutive integers that can be represented by a positive form. The discriminants I have in mind are $\Delta = -p,$ where $ p \equiv 7 \pmod 8$ is a prime with a large minimal nonresidue. Such primes can be found in particular among http://www.research.att.com/~njas/sequences/A000229 although not all of these are $\equiv 7 \pmod 8.$ The conjecture, to be more specific, is that for any of these desirable discriminants, there is a represented set of consecutive integers of length $N,$ where $N$ is the smallest quadratic nonresidue $\pmod p. $
Now, I admit I do not have any sequence of length 12 or 13 or 14. But, as with Jodie Foster in "Contact," I am the scientific type who comes around to depending on faith by the end of the movie.