We test for two critical phenomena in Amazonian ecosystems: self-organized criticality (SOC) and critical transitions. SOC is often presented in the complex systems literature as a general explanation for scale invariance in nature. In particular, this mechanism is claimed to underlie the macroscopic structure and dynamics of terrestrial ecosystems. These would be inextricably linked to the action of fire, which is conceived as an endogenous ecological process. We show that Amazonian savanna fires display the scale-invariant features characteristic of SOC but do not display SOC. The same is true in Amazonian rainforests subject to moderate drought. These findings prove that there are other causes of scale invariance in ecosystems. In contrast, we do find evidence of a critical transition to a megafire regime under extreme drought in rainforests; this phenomenon is likely to determine the time scale of a possible loss of Amazonian rainforest caused by climate change.