  We use a Metropolis algorithm to calculate the finite temperature spectral
weight of three related models that have identical quasiparticles at $T=0$, if
the exchange favors the appearance of a ferromagnetic background. The
low-energy behavior of two of the models remains equivalent at finite
temperature, however that of the third does not because its low-energy behavior
is controlled by rare events due to thermal fluctuations, which transfer
spectral weight well below the $T=0$ quasiparticle peaks and generate a
pseudogap-like phenomenology. Our results demonstrate that having $T=0$ spectra
with similar quasiparticles is not a sufficient condition to ensure that two
models are equivalent, {\em i.e.} that their low-energy properties are similar.
We also argue that the pseudogap-like phenomenology is quite generic for models
of $t$-$J$ type, appearing in any dimension and for carriers injected into both
ferromagnetic and antiferromagnetic backgrounds.
