It cannot be true: think of 2 \times 2 symmetric matrix [1           1+\eps

                                                                                                1+\esp    1]

for any \eps \geq  0 ....   

It is not easy to get an auto-correlation matrix you specified with a random process.

You are right: essentially, I offered you almost singular matrix, which is due to numerics indefinite. The simplest way of generating numerically difficult problems is to introduce deterministic linear dependence between entries of modeled multivariate variable.