I doubt this is the case. In my experience it comes down to this:
There will be administrator (God) permissions on the company instance that your personal account does not have rights to yet - If your company uses AD it may be an AD group your account belongs to that is denying rights. Comb through all the permission setting with an alternative top level administration account and you will find the blocking setting.
Alternatively you may not realise but an administrator has set your account/a parent account your account is inheriting from to deny permission to administering/change/delete process templates - check this is not the case.
VSTS permissions are very granular, accounts and groups do inherit permissions from other groups so one needs to be careful....