When you recite Tehilim, your own breath arouses the Holy Breath in these words. Therefore, when you recite Tehilim, it is as if Dovid Hamelech himself is reciting it.

When can you say Tehillim?

On most days it is before bed or maybe not at all. For Chabad, the prohibition against saying Tehillim is anytime before Chatzos. However, saying Tehillim at the bedside of the sick is OK and even commendable.

Can you say Tehillim on Shabbos?

Certainly it's admirable and desirable to say Tehillim on Shabbos. In fact, there are quite a number of Tehillim groups meeting regularly on Shabbos to complete all 5 sfarim.