In highly-interconnected timetables or dense railway traffic, a single delayed train may cause a domino effect of secondary delays over the entire network, which is a main concern to planners and dispatchers. This paper describes a stability theory to analyse timetables on sensitivity and robustness to delays based on a linear system description of a railway timetable in max-plus algebra. The max-plus model includes train interdependencies resulting from the timetable, logistics, and the shared infrastructure. Stability is the self-regulatory behaviour of the railway system to return to the steady state of the railway timetable after disruptions. The proposed approach evaluates timetable realizability and stability using max-plus spectral analysis and quantifies robustness using critical path algorithms. Moreover, delay propagation of initial delay scenarios over time and space is effectively computed by explicit recursive equations taking into account zero-order dynamics. The max-plus approach enables a real-time analysis of large-scale periodic railway timetables. A case-study of the Dutch national railway timetable illustrates the potential of the developed methodology to support the design of reliable railway timetables in dense railway traffic networks.