When gradient of utility function is a zero vector In Advanced Microeconomic theory by Jehle and Reny is said that if $\mathbf{x^*}$ is a solution to the following maximization problem $\max_{\mathbf{x} \in \mathbb{R}_+^n} u(\mathbf{x}) $  subject to $\mathbf{p \cdot x}\le y$, then $\bigtriangledown u(\mathbf{x^*})=\mathbf{0}$
is possible but quite unlikely.

The question is why is it quite unlikely? I can think only budget constraint, but is it right?
This concerns the partial derivatives of the utility function with respect to goods, and not the partial derivative of the Lagrangian of the maximization problem. 

So a zero derivative, and moreover at the optimum, would imply a threshold quantity after which utility diminishes. 

In the real world, we all know that consuming excessively may result in utility reduction (think eating too much food too quickly).  

In the theoretical world such utility functional forms have been used, especially "quadratic utility" in intertemporal representative consumer models in macroeconomics with a single good, 

$$u(x) =ax - bx^2$$

In microeconomics, all abstract development of the theory usually assumes that utility is non-decreasing in each good separately. But in cases where one would want to allow for decreasing utility, then, indeed, the reason why we would not expect to find the optimum at the zero-gradient point, would be the workings of the budget constraint together with relative prices.