  We present numerical evidence for the spontaneous breaking of the centre
symmetry of four-dimensional twisted Eguchi-Kawai models with SU(N) gauge group
and symmetric twist, for sufficiently large N. We find that for N greater or
equal than 100 this occurs for a wide range of bare couplings. Moreover for N
less or equal than 144, where we have been able to perform detailed
calculations, there is no window of couplings where the physically interesting
confined and deconfined phases appear in the reduced model. We provide a
possible interpretation for this in terms of generalised 'fluxon'
configurations. We discuss the implications of our findings for the validity
and utility of space-time reduced models as N goes to infinity.
