I had this problem when I was in a bit of a hurry to install the improperly signed driver of an instrument (the signature was accepted by older Windows version but not since 8.1 or so). I ran some trivial checks like CHKDSK, DISM and SFC (to check for filesystem and/or system-file errors) while I searched the net for alternative solutions but I had no luck. I tried to force an "advanced startup" through an admin command prompt and I realized that BCDEDIT can't touch the EFI boot manager parameters. The error was something like "can't update the boot database because some device is missing from your system". The Disk Manager indicated a healthy EFI partition and CHKDSK a healthy filesystem. So, I figured the System might have accidentally locked itself out from the protected files (or at least it thinks it did, I had no better idea or real answers). So, I quickly created an EFI-bootable Win10 USB thumbdrive and Secure Erased the SSD (fortunately, this laptop had nothing to be backed up and very little to install in order to get this particular job done but time was running out...). The result was shocking: The installer stopped with the same error message (boot code can't be updated, or something like that...). All this on a fully erased SSD. I figured it must be the fault of said SSD. I still had a little time, so as a last resort, I created a BIOS(/UEFI-CMS) bootable thumbdrive, reset the BIOS/UEFI settings of the laptop (the Setup was password protected and I checked it for accidental misconfig earlier) and tried a clean install for the last time. And it worked. I finished the job and reinstalled Win10 in EFI boot mode when I got home. I figured it must have been a problem with the laptop's UEFI firmware (some rare bug) which might or might not had anything to do with Windows 10. Now I think this is somehow liked to Win10, even if it's actually a mainboard firmware bug (which just wasn't triggered earlier). This was a HP ProBook 4535s with v60 BIOS/UEFI By the way, I am not ever sure if it's an EFI-compatible BIOS or a BIOS-compatible UEFI . It's less clear than with my desktop mainboards but I know both exists, Gigabyte calls the former hybrid-efi for example (a BIOS which can execute EFI boot managers via a special BIOS option ROM). The Setup menu is a fully UEFI-like graphical thing, although neither of the Windows versions could ever do a "reboot to UEFI Setup" on this machine (I can do that from the troubleshooter menu on my desktop PC, not on my laptop and it was always like that, even before this bug and ever since I have it, starting from Win 8.0 or even Win 7).
Did you find the cause and/or a fix? Please, let me know in case I might encounter this problem again.