This study investigates temporal suppression of click-evoked otoacoustic emissions (CEOAEs), occurring when a suppressor-click is presented close in time to a test-click (e.g. 0-8ms). Various temporal suppression methods for examining temporal changes in cochlear compression were evaluated and measured here for seven subjects, both for short- and long-latency CEOAEs. Long-latency CEOAEs (duration >20ms) typically indicate the presence of synchronised spontaneous otoacoustic emissions (SSOAEs). Temporal suppression can only be linked to changes in CEOAE-compression if the suppressor-click affects the CEOAE magnitude. Phase changes induced by the suppressor-click were shown to bias suppression in two ways: (i) when a specific asymmetric measurement method was used and (ii) when synchronisation between the CEOAE and the click-stimuli was incomplete. When such biases were eliminated, temporal suppression and augmentation (the opposite effect) were observed and shown to be subject-dependent. This indicates that the nonlinearity underlying temporal suppression can work in a more (i.e., suppressed) or less (i.e., augmented) compressive state, depending on the inter-click interval and the subject under test. Temporal suppression was shown to be comparable for CEOAEs and SSOAEs, indicating similar underlying cochlear nonlinear mechanisms. This study contributes to a better understanding of the temporal properties of cochlear dynamics.