We examine the specialization to simple matroids of certain problems in extremal matroid
theory that are concerned with bounded cocircuit size. Assume that each cocircuit of a
simple matroid M has at most d elements. We show that if M has rank 3, then M has at
most d + [lfloor ]√d[rfloor ] + 1 points, and we classify the rank-3 simple matroids M
that have exactly d + [lfloor ]√d[rfloor ] points. We show that if M is a connected
matroid of rank 4 and d is q3 with q > 1, then M has at most
q3 + q2 + q + 1 points; this upper bound is strict unless q is a
prime power, in which case the only such matroid with exactly
q3 + q2 + q + 1 points is
the projective geometry PG(3, q). We also show that if d is q4
for a positive integer q and if M has rank 5 and is vertically 5-connected, then M has at most
q4 + q3 + q2 + q + 1 points; this upper bound is
strict unless q is a prime power, in which case PG(4, q) is the
only such matroid that attains this bound.