I'm not sure why you would start with the assumption that b doesn't exist.
Regardless of whether or not the relationship between Y and the X variables is linear, it is usually possible to find a value for the vector b (exceptions might be related to multi-collinearity and non-invertibility of X'X).
This is "just" linear algebra: we are finding a set of values such that Y is projected onto the "column space" of X in a way that minimizes the error. A good reference on this is Agresti's Foundations of linear and generalized linear models
So in summary, even if b existed and was calculated, you can't immediately conclude that a linear model is the best option here. On the other hand, if b can't be calculated, you might have to drop variables from X before fitting and trying to assess whether the relationship is linear.