  We establish upper bounds on the diameter of compact K\"ahler manifolds
endowed with K\"ahler metrics whose volume form satisfies an Orlicz
integrability condition. Our results extend previous estimates due to
Fu-Guo-Song, Y.Li, and Guo-Phong-Song-Sturm. In particular, they do not involve
any constraint on the vanishing of the volume form. Moreover, we show that
singular K\"ahler-Einstein currents have finite diameter, provided that their
local potentials are H\"older continuous.
