We present various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic $$n$$ n -dimensional polytope with $$v$$ v vertices is at least $$(nv)^{\frac{1}{4}}$$ ( n v ) 1 4 improving on previous lower bounds. For polygons with $$v$$ v vertices, we show that psd rank cannot exceed $$4 \left\lceil v/6 \right\rceil $$ 4 v / 6 which in turn shows that the psd rank of a $$p \times q$$ p × q matrix of rank three is at most $$4\left\lceil \min \{p,q\}/6 \right\rceil $$ 4 min { p , q } / 6 . In general, a nonnegative matrix of rank $${k+1 \atopwithdelims ()2}$$ k + 1 2 has psd rank at least $$k$$ k and we pose the problem of deciding whether the psd rank is exactly $$k$$ k . Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when $$k$$ k is fixed.