Pros
Fellowships are typically quite stable as they are not reliant on a grant. Lots of training opportunities and support for career services. Opportunity to work with many of the world's best scientists. Ability to undertake high risk projects. Abundant resources available to allow you to excel if you grasp the opportunity. Excellent health plan for fellows.
Cons
Increasing interference by government bureaucracy affects simple daily tasks including placing orders, booking travel and even providing coffee at an NIH run event. Can feel overwhelming because it is such a large institute, you are constantly made aware of the huge number of other trainees that you will be competing with for a job. For most people, there will be little opportunity to move up within NIH as there are limited full time employee slots allotted to each lab although there are some ways around it. Large disparity in pay depending on the institute.