Yes. A simple example is
$$\max x_1 + x_2$$
$$ x_1 + x_2 \leq 1,$$
$$ x_1, x_2 \geq 0.$$
The level curves of the objective function are parallel to the boundary of the constraint $x_1 + x_2 \leq 1$, and so every point on the line segment $x_1 + x_2 = 1$ with $x_1, x_2 \geq 0$ is optimal.
This example gives you the general case: You can have multiple, nondegenerate optimal solutions if the level surfaces of the objective function are parallel to one of the constraint inequalities.
There's an interesting connection with the dual problem, too. If the original problem has multiple, nondegenerate optimal solutions, then the dual problem must have a unique, degenerate optimal solution.