This paper focuses on a representation of system reliability in the framework of possibility theory. Particularly, given a (probabilistic) quantitative knowledge pertaining to the time to failure of a system (risk function) and some qualitative knowledge about the degree of pessimism and optimism of the information supplied by the quantitative knowledge, one constructs a possibilistic reliability function of the system. The latter models the possibility of the system to survive up to the current time t. The proposal is motivated by the observation that the system may fail even if it is considered unlikely by the (probabilistic) risk function. Besides, research from cognitive science shows that probability values, particularly when a human factor is involved, tend to be overestimated or underestimated depending on the level of confidence the user is facing. This methodology involves implicitly the combination of the two pieces of knowledge into a more refined knowledge expressed in the possibility framework. Rational assumptions are put forward in order to guide the construction of the possibilistic model. Several ramifications of the proposal, particularly considering special exponential lifetime distributions will be investigated. Particularly, when reasoning in average, or looking for typical elements, new results on (possibilistic) mean time to failure (PMTF) will be pointed out. On the other hand the absence of any qualitative knowledge makes the proposal as a counterpart of the so-called probability-possibility transformations investigated in fuzzy/possibility literature. Comparisons with the preceding will be investigated for special cases of exponential lifetime distributions.