I guess part of the question would be what is the planned purpose of the BOB? Is the plan for the bag to be used when you definitely need to sleep outdoors or is it more of a low-probability event that you are planning for?

While I don't really have a BOB (so maybe take this with a grain of salt), I think this is a good way to look at it.

For example, if I'm backpacking I will choose the tent over the tarp. I prefer the comfort, weather protection and privacy it affords.

If, however, I am not planning on staying overnight when I head to the woods I would bring a tarp (or, for me, something that could be used as a tarp) as a contingency. I would not want to pack a tent for a just-in-case scenario.

So back to the BOB, if most of the events being planned for would require staying outdoors I'd lean towards including the tent if the size and weight weren't problematic. But if most of the planned events wouldn't require sleeping outside I'd likely go with a tarp or equivalent (those space blankets I linked to are a great, multi-use item that fit this bill without breaking the bank; they are about $15).