Why do Buddhists believe that the Buddha was enlightened, despite that he claimed it himself? All the answers to the question about the living enlightened people suggest that talking about one's enlightenment is taboo, so if somebody claims to be enlightened, he probably is not.

Why doesn't this reasoning apply to the Buddha? After all, he claimed himself to be englightened (for example, in Dhammacakkappavattana Sutta).
I said "bragging of one's achievements is bad tone" and you turned it into "talking about one's enlightenment is taboo" :) It is very easy to fall into extremes, isn't it?

"If somebody claims to be enlightened, he probably is not" -- here is a simple explanation. Among other things Enlightenment involves shedding the mistaken identification with the illusory "I". So once Enlightenment has been revealed, it seems rather silly to say "I am Enlightened" -- because this "I" is a fiction.

Plus, Enlightenment is self-existing, it does not depend on being attained or not. The metaphor for this is sun above the clouds. Once you remove the clouds, you can see the sun, but the sun is always there, before and after. This is another reason why saying "I have attained Enlightenment" is silly.

As Chogyam Trungpa Rinpoche said:


  Buddhism says that confusion has to be dispelled and that there should
  be realization beyond confusion. It is based on the idea of
  transcending the highest ideals of the human mind completely, which is
  called enlightenment. Once you dissolve, once your particular
  expectations dissolve, that is liberation or enlightenment. Since
  enlightenment is based on dissolving the ego and its expectations, it
  has been said that you cannot watch your own burial, and you cannot
  congratulate yourself on becoming the first buddha of the age or the
  first buddha of New York.


Buddha claimed Enlightenment solely in order to help his students gain confidence in his teaching. He knew very well the extent to which the statement "I am Enlightened Buddha" can be true.

Then why don't modern teachers claim Enlightenment to help their students gain confidence? As I explained in Is Satori the same as Enlightenment? and Enlightenment vs nibbana/nirvana, most people achieve "sudden insight / gradual cultivation". With gradual cultivation, you can never claim 100% Enlightenment, because in this case Enlightenment is asymptotic. All we can definitely claim is the insight.