I wonder how strong the power of Tannaka philosophy is, and if we accept that a tensor category is a generalized bialgebra, what difficulties we will come up against ?
Edit: Whether most tensor categories are representable, or whether for every "good enough" tensor category there exist a bialgebra with its module category isomorphic to the given category?
I'd like to explain Bruce's answer a bit more. The fusion categories Bruce mentioned have non-integer Frobenius-Perron dimensions, so it is very easy to see that they are not categories of finite dimensional modules over a bialgebra. E.g. one of the simplest of them, the so called Yang-Lie category, has simple objects $1,X$ with $X^2=X+1$. So if $X$ were a finite dimensional representation of a bialgebra, then the dimension of $X$ would be the golden ratio, which is absurd.
This, however, can be fixed if we allow weak bialgebras and weak Hopf algebras. In fact, any fusion category is the category of modules over a finite dimensional weak Hopf algebra, see arXiv.math/0203060.
As to Akhil's example (Deligne's categories), it is also true that they cannot be realized as categories of finite dimensional representations of a bialgebra (or even a weak bialgebra), but for a different reason. Namely, if X is a finite dimensional representation of a bialgebra, then the length of the object $X^{\otimes n}$ is at most ${\rm dim}(X)^n$, where ${\rm dim}$ means the vector space dimension. But in Deligne's categories, the length of $X^{\otimes n}$ grows faster as $n\to \infty$. Actually, in another paper, Deligne shows that if in a symmetric rigid tensor category over an algebraically closed field of characteristic zero, the length of $X^{\otimes n}$ grows at most exponentially, then this is the category of representations of a proalgebraic supergroup, where some fixed central order 2 element acts by parity (so essentially this IS the category of (co)modules over a bialgebra). This is, however, violently false in characteristic $p$, since if the root of unity $q$ is of order $p$, where $p$ is a prime, the the fusion categories for $U_q({\mathfrak g})$ mentioned by Bruce admit good reduction to characteristic $p$, which are semisimple symmetric rigid tensor categories with finitely many simple objects and non-integer Frobenius-Perron dimensions.
A third very simple example of a tensor category not coming from a bialgebra is the category of vector spaces graded by a finite group $G$ with associator defined by a nontrivial $3$-cocycle. This category, however, is the category of representatins of a quasibialgebra (and also of a weak bialgebra, as mentioned above).
So the conclusion is as in the previous two answers: tensor categories are more general than bialgebras. More precisely, the existence of a bialgebra for a tensor category is equivalent to the existence of a fiber functor to vector spaces, which is an additional structure that does not always exist. And if it exists, it is often not unique, so you may have many different bialgebras giving rise to the same tensor category.
The basic example is to take finite dimensional representations of $U_q(sl(2))$ and then put q a root of unity. Then quotient by the ideal of "invisible morphisms" (where f is invisible if Tr(fg)=0 for all g). This is semisimple, abelian, braided, spherical and has a finitely many isomorphism classes of simple modules.
Of course you can then repeat for $U_q(g)$ for g a semisimple Lie algebra.
Even more generally you can take the fusion category of a rational conformal field theory.
These are the examples that have been most studied and which led people to regard tensor categories as more general than bialgebras.
Edit The example given here of Deligne's categories is a valid example of tensor categories that do not arise from a generalized bialgebra. However, for the correct reason, see Pavel Etingof's answer above.
An example is Deligne's Rep(S_t) for t not an integer; this is a semisimple symmetric tensor category, but it contains an object (the "regular representation" in non-integral rank) of dimension $t$. There is a blog post by David Speyer on the subject.
There is actually a more natural way of getting a tensor category with objects of non-integral dimension (that was also known earlier); this is $Rep(GL_t)$ for $t$ not an integer. It is also covered in Deligne's paper (and cf. this post by Noah Snyder). This category is something like "the free pseudo-abelian tensor category containing an object of dimension $t$", in some 2-categorical sense: any such tensor category admits a tensor functor from $Rep(GL_t)$.
(Edited in response to comments below)
