Possingham received his Bachelor's degree with Honours in 1984, from the department of Applied Mathematics at the University of Adelaide. He completed his D.Phil. at Oxford University under Michael Bulmer in 1987, on a Rhodes Scholarship. His thesis focused on optimal foraging theory.

Possingham's first postdoctoral position was with Joan Roughgarden at Stanford University, working on the recruitment dynamics of intertidal communities. He then returned to Australia on a QEII Fellowship at the Australian National University, and undertook research on applications of population viability analysis to conservation biology. He moved to the University of Adelaide, first as a lecturer, then in 1995 as a Professor. In 2000 Possingham moved to a chair in the departments of Mathematics and Biological Sciences at the University of Queensland in Brisbane, where he was an ARC Professorial, Federation, and Laureate Fellow. He was Director of the Australian Research Council Centre of Excellence for Environmental Decisions, and the Australian government's Threatened Species Recovery Hub. In 2016 he joined The Nature Conservancy as its Chief Scientist.