Time perception is not always veridical, but it can be modulated by changes in internal and external context. The most-acknowledged theory in this regard hypothesises the existence of an internal clock allowing us to subjectively estimate time intervals. The aim of the present study is to investigate the possible effect of such an internal clock, measured as the ability to reproduce a target duration, in the mental manipulation of time: 63 healthy participants were asked to Bisect and to Double reference time intervals, besides Reproducing them. Moreover, to investigate whether time processing might be predicted by individual differences, handedness, anxiety, and personality traits were also assessed by means of standardized questionnaires. Results show that participants correctly Reproduce time intervals (internal clock), but they overestimate time intervals during Bisection and underestimate them during Doubling. We explain this unexpected pattern of results as a kind of aftereffect, due to the short-term retention (adaptation) to the subjective representation of shorter (Bisection) vs longer (Doubling) intervals, respectively. Moreover, hierarchic regression models reveal that some personality traits can predict Bisection accuracy, but they clearly show that the best predictor for both Bisection and Doubling is the accuracy in Reproducing time intervals, confirming the fundamental role of the internal clock in time estimation. We conclude that time estimation is a unique skill, mostly independent from inter-individual differences, and the new paradigms introduced here (bisection vs doubling) reveal that the correct functioning of the internal clock also explains the ability to mentally manipulate the time.