Is it possible for a system of two linear inequalities to have no solution?

Sure. Visualize the graphs of two half-planes, each representing a linear inequality. Those can overlap, or they might not overlap. For example:

x > 2, and

x < 0

But a similar example can be made with two variables, as well.

x + y > 3

x + y < 2

If you graph it, you will get two half-planes that don't touch.

If you look at the equations, for any combination of values for x and y, the result can't be both more than 3 and less than 2, so there is not a single solution.