Shanice06 the total number of tets is meaningless without any context.
How small are they relative to the frequency of deformation that you'd like to achieve?
The images I've posted above show the effect that the size of your tets will have on your deformations.
The 1st image shows the cylinder in it's rest configuration (with large tets)
The 2nd image shows the deformation that can be expressed given the provided tet mesh resolution. (in this case, the tet mesh is not high enough resolution to be able to produce the expected dynamics given the provided material properties)
The 3rd image shows a higher resolution tet mesh on the cylinder - this is now able to produce the expected deformations.
The 4th image shows the use of tet mesh spatial adaptability - in this case, the cylinder is deforming better than in the 2nd image, but not as well as in the 3rd - this is because I don't have enough elements where they are needed (closer to the side of the cylinder that is attached to the cube)
In your case, if you were to imagine that this cylinder is your "fat" layer, if the tets are too large in the areas where you need lots of deformation (especially around the joints), your tet mesh will lock, sue to insufficient degrees of freedom on the tet mesh.
I hope this helps!