It can. Plex has no room to create thumbnails, download metadata, cache for transcoding, etc.
To add to @trumpy81’s comments, you really, really need to sort out your libraries. Do not have Plex scanning items such as backups of Blu-ray discs or the recycle bin(see log entries below).
For the latter, an easy fix is to not enable the recycle bin in /volume1/Film and similar folders. You can change this in Control Panel → Shared Folder (screenshot below). Note that anything you delete from a share without a recycle bin will be deleted immediately from the NAS.
If you want to keep the recycle bin, then move your media to a sub-folder of the shared folder. For example:
/Media Server <-- Shared Folder with recycle bin enabled.
/#recycle <-- created by Synology. Files deleted from subfolders moved here.
/Movies <-- Plex Library points here
/TV Shows
/Concerts
Suggest you review the Installation & Setup of Plex Media Server on Synology and the Synology FAQ.
Straighten out your Plex libraries. That alone will increase the stability of your system. Afterwards, if you’re still experiencing No space left on device messages it will be easier to troubleshoot.
From Plex Media Server.log
Plex scanning a Blu-ray backup folder:
Summary
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/PLAYLIST"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/CLIPINF"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/AUXDATA"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/META"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/META/DL"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/META/ES"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/META/TN"
Jul 04, 2020 14:17:23.498 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/STREAM"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/JAR"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/BDJO"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/BACKUP"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/BACKUP/CLIPINF"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/BACKUP/PLAYLIST"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/BDMV/BACKUP/BDJO"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/Queen"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/Queen/images"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/CERTIFICATE"
Jul 04, 2020 14:17:23.499 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Concerts/Queen Montreal/QUEEN_ROCK_MONTREAL/CERTIFICATE/BACKUP"
Plex scanning a Recycle folder (this goes on for 290 lines):
Summary
Jul 04, 2020 14:17:23.503 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle"
Jul 04, 2020 14:17:23.503 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/_SHOWS"
Jul 04, 2020 14:17:23.503 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/_SHOWS/Big Love"
Jul 04, 2020 14:17:23.503 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/_SHOWS/Big Love/Big Love Season 2"
Jul 04, 2020 14:17:23.503 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Behind Enemy Lines"
Jul 04, 2020 14:17:23.503 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/The Secret"
Jul 04, 2020 14:17:23.504 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/John Wooden Ucla Offense"
Jul 04, 2020 14:17:23.504 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Erin.Brockovich.2000.Bluray.AVCHD.720p.x264.ac3.HD-R-US"
Jul 04, 2020 14:17:23.504 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Erin.Brockovich.2000.Bluray.AVCHD.720p.x264.ac3.HD-R-US/BDMV"
Jul 04, 2020 14:17:23.504 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Erin.Brockovich.2000.Bluray.AVCHD.720p.x264.ac3.HD-R-US/BDMV/PLAYLIST"
Jul 04, 2020 14:17:23.504 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Erin.Brockovich.2000.Bluray.AVCHD.720p.x264.ac3.HD-R-US/BDMV/BACKUP"
Jul 04, 2020 14:17:23.504 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Erin.Brockovich.2000.Bluray.AVCHD.720p.x264.ac3.HD-R-US/BDMV/BACKUP/CLIPINF"
Jul 04, 2020 14:17:23.505 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Erin.Brockovich.2000.Bluray.AVCHD.720p.x264.ac3.HD-R-US/BDMV/BACKUP/PLAYLIST"
Jul 04, 2020 14:17:23.505 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Erin.Brockovich.2000.Bluray.AVCHD.720p.x264.ac3.HD-R-US/BDMV/CLIPINF"
Jul 04, 2020 14:17:23.505 [0x7f19924f4700] DEBUG - [Notify] Now watching "/volume1/Film/#recycle/Erin.Brockovich.2000.Bluray.AVCHD.720p.x264.ac3.HD-R-US/BDMV/STREAM"
Synology DSM:
