Linear regression when a non-zero intercept is theoretically implausible How should I think of a linear model with a positive intercept when, theoretically, the intercept has to be zero?
Think of the following example: we are modeling how many birds does a feral cat hunt daily, predicted by the number of steps it has taken, the number of leaps, and the number of times it has extended its claws. Let’s say our model’s intercept is different from zero, and positive, yet we know that if the cat didn’t take a single step, leapt a single leap, nor extended its claws, it has certainly not caught any birds.
What does a non-zero intercept tell us in the case where it defies reason? Does this indicate that our model is inaccurate? Should we drop the intercept from our model?
 A: As George Box said, "all models are wrong, but some are useful". If you've built the model described above you were presumably interested in modeling cats that have typical hunting patterns. Your theoretical cat is probably too far from the actual data to accurately predict. If most cats take 10k-20k steps (a complete guess) then extrapolating to 0 is quite far and I would question the result. If the model fits the actual data well then I wouldn't worry about these theoretical cats. If you are concerned with immobile cats, find more samples filling in the gaps down to zero. you may find that a linear fit is not appropriate as these new data are added. 
A: 
How should I think of a linear model with a positive intercept when, theoretically, the intercept has to be zero?

Computers will estimate whatever we ask of them, regardless of if it makes sense or not.  By letting the intercept be a parameter, you are saying that you don't know how many birds the cat will have hunted given its steps, and so you want to estimate it. The model will return an estimate of exactly 0 with probability 0, so relying on the model (or the computer) to make important decisions is clearly a bad idea.

What does a non-zero intercept tell us in the case where it defies reason?

It tells is that we need to rethink what we are trying to model; that we have misinterpreted something along the way and we need to go back and rectify it.
