Actually we've had this same topic only a few weeks ago, although at first the emphasis was place upon the Dutch, Portuguese, & Spanish.
Anyways... to put it simply, yes. Someone else could have as the Dutch were only a stone's throw away up in Java. In fact Able Tasman maped large parts of Australia long before Capt Cook arrived on the scene.
Likewise the Portuguese weren't far away either.
But the French. Well they were sniffing out this way too. In fact, a handful of days after the First Fleet arrived in 1788, a French expedition, lead by the famed explorer La Perouse, landed & camped at Botany Bay, which is some 15km to the south of Sydney Cove.
IMHO, though, I think it's the Dutch who could have set up shop here before any of the others if they wanted. So, decades (if not a century) before the British founded their Sydney colony, the Dutch could have already had colonies somewhere near Darwin, Perth, Adelaide, & Hobart, if not elsehwere. That'll make it rather difficult for the British to then grab the rest of the continent. And that's if the British actually wanted to do this, as at first the colony at Sydney was essentially about having a place to dump convicts more so than anything else.
Of course all this Eurocentric view misses the opportunities Asian peoples had in occupying Australia before the Europeans ventured out this way. So if we include them, in the discussion, well the Chinese of the Ming era certainly become a leading candidate, not to mention the Malays, Javanese, &, according to Ridwan Asher (who's an Indonesian AH.Com member), the Bugis.