MacBook Pro battery will not charge even though cycle count is below max count

Any Apple store or Authorized Service Provider can run a diagnostic for you within a few minutes to gauge the health of the battery for the official Apple answer to your question.

However, saving a trip and appointment and just looking at what you posted, it looks like it is in correct working order for the age.

The design capacity is the total charge the battery was produced for. If you ran this check on the day you bought the MacBook the current capacity should have been right around 5200. Just like a rechargeable battery in a phone, toy, etc. it never fully charges to its designed capacity over time. It is just the nature of rechargeable batteries. So now, your battery when fully charged will only charge to the current capacity.

Given that you have used it for 17 months. These results seem in-line.

According to this Apple KB Article, your model does not require any calibration.

They also publish some tips in this KB Article.

Here's a great app for checking the battery and it gives some great definitions and links to Apple as well.

Hope that helps.

The cycle count is logged on the battery itself.

I've never actually dug into the idea to see the method or technology used to store the cycle count, however I do know this to be true thanks to personally refurbishing hundreds of MacBooks. When checking over a MacBook's battery life in System Information / System Profiler I would often switch out batteries and check again, deciding if a battery replacement was worthwhile. Once the battery is switched out you will see the cycle count (along with capacity, etc) change. If you don't see it change then you either need to refresh System Profiler (command + R) or the second battery just happens to have the same info.

If you're curious you could always visit the Genius Bar at an Apple store and have them pop in one of their spare batteries that they use for testing.