Neocortical Layer 6, A Review (Alex M. Thomson, 2010) shows that L6a and L6b CT cell initial EPSPs are weak, so it probably takes a bit of input. It might be more accurate to say that the cell starts off firing slowly, but fires more quickly as long as the input keeps coming.
I’m not sure if there is spike facilitation, but there might be, because some cells adapt, what you could call spike depression.
I’ve only heard of synaptic facilitation, which doesn’t depend on whether or not the postsynaptic cell fires. Instead, whenever the synapse sends a signal, it is temporarily weakened. This means that repetitive inputs still cause a weakening signal even if the cell never fired.