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. I believe the prohibition involves the daily portion of Tehillim. From others (non-Chabad), I have heard that night is the time of gevurah while day is the time of chesed, and that the recitation of Tehillim at night invites judgment rather than chesed. The exceptions would be on Friday night, the 48 hours of Rosh Hashanah and on Yom Kippur.
My grandparents began Tehillim before shkiah and concluded after nightfall. This was OK because it was just a continuation of an action initiated before nightfall.