Assume I have a firm-value credit model in which the default time [$]\tau^*[$] is subject to the evolution of the assets [$]X[$]. Also assume that the default time is defined as [$]{\tau ^*} = \inf \left\{ {t \in {\mathbb{R}^ + }:X(t) \le h} \right\}[$], where [$]h[$] is the default barrier. This is a standard setup.
If [$]X[$] is a continuous process such as GBM, then [$]\tau^*[$] is predictable (I can create an increasing sequence of stopping times that announce the default). This is easy.

In the reduced-form models the default time is a total surprise and the default time is totally inaccessible (intuitively, knowing the default intensity still tells us nothing about how close to default the company is; there is no announcing sequence and we have no idea when the company defaults even when we know the intensity).

But what if I have a firm-value model where the assets [$]X[$] follow some jump diffusion (with the jump component based e.g. on Poisson). Is it possible to classify the default time [$]\tau^*[$] somehow? Here,  [$]\tau^*[$] seems to me to be somewhere between 'predictable' and 'totally inaccessible'. I can partially see the company to get closer to the default but if the assets jump, then the default is still a surprise. 

Well, it is obvious that [$]\tau^* = \min \{\tau_A,\tau_B\}[$], where A is a predictable event (here, first diffusive touch of h) and B is a totally inaccessible event (here, first jump across h). That this type of decomposition of a stopping time always exists, with A and B disjoint events, is Th 3, pg 104 in Protter's 'Stochastic Integration and Differential Eqns' (2nd edition). 

Well, it is obvious that [$]\tau^* = \min \{\tau_A,\tau_B\}[$], where A is a predictable event (here, first diffusive touch of h) and B is a totally inaccessible event (here, first jump across h). That this type of decomposition of a stopping time always exists, with A and B disjoint events, is Th 3, pg 104 in Protter's 'Stochastic Integration and Differential Eqns' (2nd edition). 

In math theory they always assume that Poisson and Wiener components of the jump-diffusion process are independent. I do not know it might exist or not conditional distribution formula for [$]\min \{\tau_A,\tau_B\}[$] given [$]\tau_A[$] or [$]\tau_B[$]. It will help to calculate distribution of their minimum.

Well, it is obvious that [$]\tau^* = \min \{\tau_A,\tau_B\}[$], where A is a predictable event (here, first diffusive touch of h) and B is a totally inaccessible event (here, first jump across h). That this type of decomposition of a stopping time always exists, with A and B disjoint events, is Th 3, pg 104 in Protter's 'Stochastic Integration and Differential Eqns' (2nd edition). 

I once saw Protter, in the setting of a math finance seminar describe a stopping time as "like, totally inaccessible, dude"

Well, it is obvious that [$]\tau^* = \min \{\tau_A,\tau_B\}[$], where A is a predictable event (here, first diffusive touch of h) and B is a totally inaccessible event (here, first jump across h). That this type of decomposition of a stopping time always exists, with A and B disjoint events, is Th 3, pg 104 in Protter's 'Stochastic Integration and Differential Eqns' (2nd edition). 

I once saw Protter, in the setting of a math finance seminar describe a stopping time as "like, totally inaccessible, dude"

Haha. I didn't know he was from the Valley.

Well, it is obvious that [$]\tau^* = \min \{\tau_A,\tau_B\}[$], where A is a predictable event (here, first diffusive touch of h) and B is a totally inaccessible event (here, first jump across h). That this type of decomposition of a stopping time always exists, with A and B disjoint events, is Th 3, pg 104 in Protter's 'Stochastic Integration and Differential Eqns' (2nd edition). 

I once saw Protter, in the setting of a math finance seminar describe a stopping time as "like, totally inaccessible, dude"

Protter probably talk a particular case which we do not know now. If we admit GBM for a stock dynamics and we are waiting for a stock will reach a particular level such as say $18 or $4 then these moments are markovian the a SDE is make sense to consider up to these moments and it is not look too stupid to observe the arriving at these events.