Updated with zero problems, running on VMware ESXi 5.5 Update 3a & ESXi 6.0 Update 1a!!
We need to understand how you are running your system, which platform? (Bare-metal, virtual machine, etc.)
You can't just run the update from the web interface if you are running DSM 5.2-5565 or else you will brick your device. You first need to update your bootloader to the latest (XPEnoboot - 5.2-5644.1) via ISO, IMG, or VMDK.
1. Boot with the new bootloader and select the 3rd option to "upgrade"
2. Launch your Synology Assistant and connect to your NAS.
3. Select "Manual Upgrade" and select the newest downloaded .pat file
4. Run the upgrade and reboot. If all is successful, you will boot back up and now be running on DSM 5.2-5644
5. Go to Control Panel>Update & Restore, and update to 5.2-5644 Update 1 (aka 5.2-5644.1)
6. Reboot - if update was successful, you will now be on the latest version and will have upgraded your device successfully.
I hope this helps!