  The dimension of a block design is the maximum positive integer $d$ such that
any $d$ of its points are contained in a proper subdesign. Pairwise balanced
designs PBD$(v,K)$ have dimension at least two as long as not all points are on
the same line. On the other hand, designs of dimension three appear to be very
scarce. We study designs of dimension three with block sizes in $K=\{3,4\}$ or
$\{3,5\}$, obtaining several explicit constructions and one nonexistence result
in the latter case. As applications, we obtain a result on dimension three
triple systems having arbitrary index as well as symmetric latin squares which
are covered in a similar sense by proper subsquares.
