Actually, the Roman Republic lasted no more than about 300 years, followed by 400+ years of empire and despotism.
From abundance to selfishness; From selfishness to apathy; From apathy to dependence; From dependence back into bondage."
While these delineations are far too neat -- they don't happen in such clear-cut phases -- we've seen enough selfishness, apathy, and dependence over the past 35+ years to surmise that republics are not immune.