We investigate a timed non-interference property for security systems modeled as timed automata, in which a low-security level user should not be able to deduce the occurrence of some high-security level actions. We assume an attack model in which the malicious (low-level) user has the ability to partially observe and memorize the set of runs of the timed automaton modeling the system. We first formalize a non-interference property that ensures the system security under such an attack model and we then prove the undecidability of that property when the attacker can have an arbitrarily big memory, i. e., when they are able to memorize sequences of previous observations, with time-stamps, of any length. We next assume bounded memory for the attacker and show that the property can then be decided in PSPACE for a subclass of timed automata ensuring finite duration between distinct observations.