There are a couple of potential problems. First, the assumption of periodic simulations is that your box size is large enough that there is no effect on an atom from its nearest periodic replica. I come from the world of biomolecular simulations, where we typically "pad" our proteins with at least 1 nm of water. I would guess that your entire proposed simulation box is not much more than 1 nm to a side, so you might get some really weird dynamics from atoms that "see" their periodic images.
Second, even if your simulation is big enough to avoid that, there can be finite size effects in the properties of the materials you're studying. That is to say, whatever observable you try to calculate may actually depend on the size of the box. (In some cases, such as diffusion in liquids, this can be used to extrapolate to the bulk limit by simulating at several sizes.) This is even a risk in the large-scale production simulations for publications, but at very small box sizes the effects will be more pronounced.
Solution: You're probably underestimating the power of your hardware. I regularly run simulations of 1500-2000 atoms on my 5 year old laptop (using molecular mechanics). It's fast enough that I can use these systems to test the methods I develop, and (given my choice of system) "real" enough that when it comes time to publish, it's considered an acceptable test of the method.
You might be interested in these old LAMMPS benchmarks: https://lammps.sandia.gov/bench.html. For example, under "metallic solid," the 1-processor tests say that a Linux desktop in 2012 ran 100 steps of 32000 atoms in under 6 seconds. The protein example is a bit slower, coming in at 36 seconds on that computer.
