Ravi - Here is a `global' example of something being representable that
shouldn't be.  Let F be the functor that sends X to isomorphism classes of
Z-torsors on X (in the etale topology and where Z=group of integers).  
These are classified by homomorphisms from the algebraic pi_1 to Z, but
there aren't any since pi_1 is pro-finite, and so all Z-torsors are
trivial.  Therefore, F is represented by a point, but that's different
from the stack BZ since the trivial Z-torsor has automorphism group Z.
I'll admit that's it's a little artificial, but it's also much less
pathological than I would have thought necessary. - Jim

Jan. 25, 2006:  Brian Osserman sends this follow-up:

Hi Ravi,
   I noticed the example you had posted from Jim Borger at
http://math.stanford.edu/~vakil/727/jim
(incidentally, this is the one I was trying to remember last year when I
asked about stacks where the functor is representable but the stack has
automorphisms), and it is related to something Max and I are talking about
now.
   I discussed it with him, and we decided that the example is incorrect.
The argument is wrong because that classification of torsors only works for
finite groups in the case of the algebraic fundamental groups (i.e., it
is too good to be true). Moreover, one can construct counterexamples 
fairly easily: eg, take the trivial Z-torsor on P^1, glue the base to
itself to get a nodal curve, and glue each component of the torsor to
the next one. 
   Max thinks he can prove that there are no Z-torsors on any X which is
geometrically unibranch, so I think this says that one gets examples
along the lines Jim had in mind for stacks of Z-torsors on the small
etale site of a lot of schemes (but I guess this isn't nearly as
interesting).
   Best,
  Brian


Later that same day, from Max Lieblich:

Let me record the proof in question: suppose X is (irreducible and)
geometrically unibranch.  Let eta be the generic point, and let i:eta -> X
be the inclusion.  It is easy to check that i_* Z = Z (because X is geom.
unib.).  Now look at the Leray spectral sequence: you end up with an
injection H^1(X,i_* Z) -> H^1(eta,Z).  But H^1(eta,Z) is Galois cohomology,
and it is easy to see that this must vanish.  E.g., H^1(eta,Z) is the
colimit of the finite Galois cohomologies, so it is indeed continuous maps
from the (profinite) Galois group of eta to the (discrete) group Z, hence
vanishes.  Another way to see the vanishing of the Galois cohomology is to
look at the sequence 0 -> Z -> Q -> Q/Z -> 0; you see that H^1(Z) injects
into H^1(Q), which vanishes because Q is uniquely divisible and the Galois
cohomology is torsion (so that any Galois cocycle can be explicitly realized
as a coboundary).

Max

From Jim:  

I would like to add a comment about that 'large' (or whatever)
fundamental group, which is Z for the nodal P^1 but probably agrees
with the etale fundamental group in the geometrically unibranch case.
I'm pretty sure there is an SGA expose about it, but maybe it's one of
the missing ones or something.  Do you know anything about this? 
 -- Jim