The first discovery this type of symmetry (in synthetic Al-Mn alloys in 1982) led to the 2011 Nobel Prize in chemistry 
Crystals require translational symmetry, therefore, no, not in 3-D space.
In 4-D you can have 5-fold symmetry, and a projection of these onto 3D space makes an arrangement with 5-fold symmetry which make quasi-crystals.
My own experience:
These have been observed in nature in nanoparticles, and are even the energy lowest state (!!). As they grow, however, there is a an energy strain penalty due to lacking translational symmetry that destabilizes them,.
Mathematically, yes, but atoms are on a discreet lattice locations. If you used time for the fourth dimension, you'd have to jump in time discreetly. That's kinda weird because where would the energy go? A photon? Then how do you recapture the energy? Do you "borrow" the energy (exploiting some uncertainty principle)? These are all high energy processes -> not a crystal (crystals are low energy states).
I’d be interested to see a simulation that’s able to make a time crystal that can have symmetry between spatial and time dimensions, even if the physics are tuned for them rather than our universe. Just to see what it would look like. I’d take any new imagery to aid in understanding solid state physics.
This is where it starts getting interesting. Where is the "location" of an atom? Is it the nucleus? Is it the electrons? Is it just the electrons which are localized to particular nuclei? Etc.
Typically the "location" has some uncertainty, both from thermal jitter and from the nature of orbitals as probability densities.