Can the Bourbaki series be used profitably by undergraduates?
1) No, you are not the target audience.
2) No undergraduate is.
3) For every subject an undergraduate might want to learn there are books better suited than Bourbaki: ask about them here.
4) In a strictly historic sense the target audience was Henri Cartan and the aim of Bourbaki was to give him a text from which he could teach Stokes' theorem rigorously, so that he would stop badgering André Weil. (Bourbaki had estimated in 1934 that it would take six months to write the book)
5) That aim has never been achieved.
6) The thousands of pages written by Bourbaki started as prerequisites for the proof of Stokes' theorems. Then they decided to forget about Stokes and write a treatise containing all of basic mathematics. They failed to achieve that goal too. (There is for example an allusion to a forthcoming book on numerical mathematics...)
7) At the graduate level certain books by Bourbaki are still arguably the best available references: some volumes in Algebra, Commutative Algebra and Lie Theory come to mind.
8) Bourbaki is not dead: they have published in 2012 a completely modified new version of Algèbre, Chapitre 8, about 3 times the size of the old version, published in 1958. (Recall that in 1934 they had estimated that the whole treatise would have been written within 6 months...)
Edit
Bourbaki is less and less dead: in 2016 they published the fat volume
Topologie Algébrique: Chapitres 1 à 4.
Here is a preview on Google Books (click on Aperçu du livre, just below the first image).
I would like to provide a slightly different answer to the question "Can the Bourbaki series be used profitably by undergraduates and high school students?" from personal experience.
I personally have read most of the Bourbaki volumes. I just finished my undergraduate education this past semester. However, I was fairly advanced as an undergraduate, and I took the graduate sequences at my school in a number of subjects.
I think that I learned a lot from Bourbaki, but it was almost never the first source that I went to. The only exception was the first volume on Set Theory. Because it is listed as a prerequisite for the later volumes, I read through it despite having only the basic undergraduate knowledge of set theory. This was fairly difficult, and in retrospect I would have been better off reading a more elementary book first on the subject. It is also not as much of a prerequisites as the writers would like you to believe; you really only need elementary undergraduate set theory to understand most of the later books.
As for the other volumes, I read all of them except the volume on spectral theory and about half of the volume on commutative algebra. None of them were easy reads. I did not read the algebra and topology ones until after I had taken graduate classes on the subjects, and the other ones after I had at least read another book on the same subject. I doubt I would have gotten anything out of them before that point. The easiest for me personally was topology, but this probably varies a lot from person to person. In most cases, the Bourbaki books were the last, rather than the first, introductory books I read (introductory here should be taken to mean that the exposition was relatively complete with few formal prerequisites).
So my answer is that the Bourbaki books can be used profitably by undergraduates, but only those undergraduates who are fairly advanced. If you're at the level of learning the things for the first time (where most undergraduates are), or you're expecting an easy read, then there are much better books. And the target audience is most definitely not undergraduates.
Just to make things clear, I love Bourbaki.
However... These books are not made to teach, they aim rather at generality and exhaustivity. And they succeed very well I think. Some volumes are getting old fashioned, like Théorie des ensembles, or Intégration. Algèbre and Algèbre commutative are just great, I often need a generality I can only find there.
Concerning the rigor, they are rigorous in the narrow meaning of the term : they will always prefer generality and precision rather than readability and ease of use. But in the other hand, the logical frame is clear, references are precise, you know where are the definition, you know what they mean, always. They are rigorist.
With Bourbaki, all starts from the beginning. You might think that you can learn all math simply reading Bourbaki, but that just wrong. And they know that very well.
In the end, if your question is really “are you the target audience ?”, the answer is no. If the question is “is it interesting for me ?”, then you should read it, and you will see. Bourbaki is one of the most influential mathematician (fictional, but still) of the 20th century, and also one of the most controversial (is formalism a dead end ?), you should know what is it like, what is his philosophy, and have your opinion on it.
My advice is to read first the introduction of Théorie des ensembles which explains the general philosophy, and then read the volume about the field you know the best.