Well, there are several bursts in species diversity I can think of.
The Cambrian and Ordovician explosions resulted in a vast increase
in animal diversity. Likewise, after the one-two punch of the
Permian and Triassic extinctions, the number of marine animals
rose steadily (**) to an all-time high (*) just prior to the spread
of humans.

(**) biggest exception being the K/T (bye bye dinos) extinction

(*) about 800 families

Also, plants arose from green algae and colonized the land in
succesive sweeps. Mosses colonized very wet environments first,
ferns (who had evolved vascular tissues) took over more territory
when they evolved (1). These were eventually (mostly) replaced by gymnosperms
(pines and the like) (2) and then (mostly) displaced by angiosperms (flowering
plants -- now the dominant plant group on the planet(3). Fungi
also radiated greatly with the invasion of the land.

(1) around the carboniferous (up to about 200 families)
(2) around the triassic (up to maybe 250 families)
(3) starting in the cretaceous (rising to about 600 families currently)

It's unclear (to me at least) what the max equilibrium number
of species the earth can hold (***) and if it has ever hit this
in the past. It could be (warning: speculation alert) that
diversity has never reached a peak because mass extinctions
happen often enough to keep the total number down.

(***) This would depend a great deal on how fragmented
specific ecosystems were.

See Cowen's book "History of Life" for a not-too-technical
run-down on, well, like the title sez, the history of life.
Or see, Wilson's "Diversity of Life" for a view centered more
on current ecology -- this is (IMHO) the best popular biology
book of (what the hell, I'll say it) all time.

Chris Colby   ---      email: colby@bu-bio.bu.edu     ---
"'My boy,' he said, 'you are descended from a long line of determined,
resourceful, microscopic tadpoles--champions every one.'"
      --Kurt Vonnegut from "Galapagos"