If we're talking in the core of the term's meaning, that a nation is notable and extraordinary in some way, then it's up to them.
All the great powers of the world can shout it from the mountain tops (America, UK, France, China, Japan, Germany, etc), along with nations that have vibrant cultures and and are rising (like Brazil or India, for example) or "small", prosperous nations like Norway or Denmark. Pride is good in nice doses and a feeling of "special-ness" helps a nation long as these things don't translate to "we're better than you and you are an inferior people for not being on our level". That's where the issues come in, America and many other nations have had this issue in some shape or form throughout history.
Far as prideful citizens go, everyone thinks their nation is the best. I've seen this with Colombians, Canadians, Americans, Brits, Italians, hell even Cubans, and so on. It's about more than economic power or test scores.
Personally, I don't think every nation is exceptional. But, I think that label is up for debate entirely and not something that's easily settled. Couldn't help but laugh when Putin said America wasn't exceptional -- a history of feats and innovation says otherwise, but it was obviously a hot case of bull****. Though, I also think Russia is an exceptional nation.
You could definitely say that there are nations exceptional in far more ways than others, no doubt. People don't need to go hanging their heads down to avoid hurting someone's feelings, we are all not on the same level when it comes to something like this. Of course, none of this changes that we should shift towards a world that doesn't **** on one another -- but that's a dream for now.