Well, in the classical theory of electricity and magnetism, no, there is no upper limit on the strength of a magnetic field.
But quantum mechanics and relativity do introduce some constraints which aren't so trivial and are worth talking about. There are limits to the strength of an electric field which extends over a macroscopic volume. Electric and magnetic fields store energy with a density proportional to their squares. In a vacuum, with no particles which have charge in it, a very strong electric field can create electron-positron pairs which fly apart, lowering the energy of the system below what it is without those e+e- pairs. The charges will screen the field until it drops below the critical strength required to make the e+e- pairs. The size of the volume over which the field is this strong is also important. If the field is strong only in a small volume, say, in the immediate vicinity of an electron (which we think is a point particle, to the best of our measurements), then there isn't enough energy stored in the field to generate a real electron-positron pair. There isn't a sharp cutoff in the field strength needed to do this, due to the fact that the e+e- pair needs a bit of quantum tunneling to work -- the e+ and e- have to separate a bit from each other in order for the system's potential energy to be less with the e+ and e- than without, taking into account the energy needed to create the e+ and e- in the first place (about one million electron volts). So there is a classical "barrier" -- the mass needs to be created, and then separate in order to justify the energy expenditure to create the mass in the first place. My guess is that if the electrons can manage to get separated by 1 fm (1E-15 meters) and get enough energy from the external field to justify their masses, you'd get plenty of spontaneous pair creation. So about one million volts per 1E-15 meters would do the trick, and the field would have to have an extent of a box several fm on a side.
Virtual e+e- pairs are constantly created and destroyed in the vicinities of real electrons, and they help to screen the charge of electrons and weaken the observed electric field.
So why am I bothering with electric fields, since you asked a question about magnetic fields? It turns out that magnetic fields in one frame of reference are combinations of electric fields and magnetic fields in other frames of reference, moving with respect to the first. There are no magnetic charges, corresponding to electric charges, to make the argument of creating pairs of them to reduce the stored energy in a magnetic field, but there is still the e+e- argument, which should work in all frames of reference.
Electrons have magnetic fields associated with them because they are charged and they spin. Because the electron is a point particle (or at least a very small particle), the magnetic field in the immediate vicinity of the electron also is arbitrarily strong over very short distances. The cloud of virtual e+e- pairs also screens this magnetic field a bit too, but it still grows to infinite strength for infinitely small electrons.
On the practical side, the strongest man-made magnets are already discussed on this site, and the limitations come from the properties of the materials and conditions under which we subject them. The strongest man-made magnets I have heard of are ones where you take an already strong electromagnet and implode it by stacking explosive materials around it. You can get a field with a strength of >1000 Tesla, but only for a few microseconds. Magnets which can sustain a constant field have to withstand the magnetic forces on the materials of which they are made, and the limits are a few tens to perhaps 100 Tesla, but people are constantly working to increase these numbers.
Strong magnetic fields can be found in the vicinity of neutron stars. Magnetic fields of order 10^11 Tesla (=10^15 Gauss) are predicted in the vicinity of magnetars, a special class of very magnetic neutron star.
(published on 10/22/2007)