It sounds like it could possibly be a problem with the 'intelligent viewfinder display". It has a direct impact on viewfinder brightness for the 7D, 1DX and 5D3. Under normal operating circumstances it shouldn't be a problem with a good battery, but all manufactured electronics are prone to some level of problem.
Here is Canon's description of the IVD.
http://www.learn.usa.canon.com …_viewfinder_display.htmlp
"The LCD overlay
Like the EOS 7D, both the EOS-1D X and EOS 5D Mark III use a transparent layer above the focus screen that can display black LCD markings as well as briefly illuminate these markings in red. Like many LCD panels, the beauty of this viewfinder display is that only the information that needs to show is visible and the finder can remain uncluttered for maximum ease of viewing the subject.
The LCD overlay does require a tiny amount of electrical power to operate. This is obviously no concern when the camera is turned on, but if the battery is removed the transmissive LCD suddenly loses a lot of brightness and contrast. This is perfectly normal and will return to full brightness once a battery is reinstalled in the camera (the camera doesn't have to be turned on; it only requires a functioning battery pack to draw power for proper viewfinder operation)."
The Intelligent Viewfinder Display is not found on: 40D, 50D, 5DC, 5D2, 1D3, 1D4 as a partial list. It is a newer introduction to Canon DSLRs.
Since it's electronic, it's entirely possible there is some sort of problem with the Intelligent Viewfinder Display.
Only other thing I could think of is a feature I'm not sure is on the 5D3: (this is from a 1D4 manual) Custom Function III-14 AF Point Brightness: 0: Normal, 1: Brighter makes the AF point light up brighter in red.
I understand the OP is talking about the green information displayed in the viewfinder (and not the AF points), but the transmissive LCD overlay and it's ability to lose brightness has always been a potential concern for over the traditional viewfinders found in most other Canon DSLRs that don't require power for normal constant and consistent brightness.