@Iridium: My NAS is simply my Windows 10 PC and the files are on an NTFS drive.
I did not have this issues in the past, they have started recently after I moved to a new home.
The WLAN router is relatively close to the RPi, and I am using a simple USB dongle (no antenna) purchased with my RPi.
I will test your suggestions to eliminate the most probable causes. The one thing that still bugs me is the power supply issue. I remember reading that RPi is extremely sensitive to power supply. Anyway to check if this is part of the problem?