For most people, I would say no, they lack the self-awareness and self-control to be able to make such a judgement. Those who have practiced a good amount of meditation would, on average, be better candidates. Moreover, nearly anyone experiencing strong emotions would have extra trouble thinking objectively and detached from personal interest. Those who can turn their emotions off would be better candidates.
Another method to consider is creating an objective measuring system that does not rely on subjective information. Technology or strict definitions may be needed for the method to be reliably applicable. If the person making the measurement system created it with the intent to use on others, it could be good to use back on the creator. Some simple examples here are those measurement systems intended to assess physical abilities and strength. Granted one had reasonable self-control and integrity, it would be pretty easy to accurately measure one's own physical strength using a weight set, for example. A fair person could also measure his or her own maximum running speed with pretty good accuracy.
As for matters of "significant vested interest", I would need some specific examples to really be able to say whether it would be reasonable to expect a person to be able to self-assess. The only example that comes to mind right now is that of whether the world would be a better place without oneself. That would be difficult for many people to answer accurately since, at least it seems to me, many people do not consider their own value to society when considering their career or other choices that affect others. Some people have a net positive effect on society while others have a net negative.