As a rule of thumb, only very strong earthquakes (like 8.0+) can produce these very large, destructive tsunamis, and an 8.0+ quake lasts a long time, so I highly doubt that you could ever miss feeling the Cascadia subduction zone letting loose in a big way.

On one of these video clips, they were interviewing someone from the Australian tsunami warning agency. He broke down the various things that need to happen between the time of a quake and when Joe Average on the ground actually receives a warning to evacuate and he says it's very difficult to do so under 30 minutes.

Although there can be automatic algorithms hooked up to sensors that will send out a warning, it does take a human time to analyze the specifics of a quake, it's location, intensity, depth, the ocean topography, whether there was ocean floor displacement, etc. to make a more definitive prediction. Which is not a good thing for the PNW. But yeah, any time you feel a really strong earthquake in the PNW--the kind where you're muttering, "Holy ****...holy ****..." and everything is getting knocked down, head to higher ground as soon as it's safe if you're near the coast and near sea level. In the Sendai quake, the tsunami pushed 6 miles inland!

However, the other possibility is that the quake happens far enough away that the intensity is a lot lower. I think that a major quake in the Aleutians, near Alaska, could also generate a large tsunami that sweeps down to the PNW without major shaking being felt in the lower 48 and get there pretty quickly.