Is there any case in which excitability increases with lowering the RMP?
A typical voltage-gated sodium channel has 3 gating states: open, closed, and inactivated. The third category is the important one here.

Open channels are open: they let ions through.

Closed channels are closed: they don't let ions through, however, they are available to open and their probability of opening increases as the membrane voltage rises from negative towards positive.

Inactivated channels don't pass ions, but it also cannot be opened by depolarized voltage. Instead, it needs a negative membrane voltage to pass back to the Closed state. 

When a sodium channel opens, it only stays that way briefly before going to an inactivated state. Recovery from inactivated back to "closed but openable" requires the 

If you depolarize a cell gradually, this will open sodium channels, but it will also increase the number of channels stuck in this "inactivated" state. That means there are fewer channels available to flip from closed to open.

When a cell is hyperpolarized, there are fewer channels stuck in the inactivated state, and so the threshold potential can actually become more negative: it takes a smaller voltage step to open enough channels to trigger an action potential, because there are more channels available to open. You can describe this situation as being "more excitable" despite the hyperpolarized potential.

Some cells, including cardiac cells and other types of pacemaker neurons, will actually depolarize spontaneously from this hyperpolarized potential and reach threshold all on their own without external input.