Applying for full-time after declining an internship position

Based on my experience, that is no issue at all. If you reject an offer (either for an internship or for a full time position), there are typically no grudges whatsoever for the future. In some cases, that can even make you more interesting.

You will need to apply again for a full time position, but you should most likely receive an invitation to the interview process.

In the interviews, you will probably receive a question why you don't go full time to McKinsey (as per your example) after your internship. You have two options:

1. Mention that it didn't work out - that's honest, especially if you reflect about it and can point out what you have learned since then and how you grew.
2. Don't mention that it didn't work out at McKinsey -- but also, don't lie about it. Just say that you want to explore your options before signing on full time for any firm, which is totally fair to do.

As Achyut already pointed out, don't be negative ("condescending or insulting") about the firm where it didn't work out. That's considered really bad style and would definitely work against you.

Good luck and all the best!

It might vary from office to office, but I think this should be no problem if you are honest and transparent with the other firms (especially about the reason you're choosing firm X over Y). As long as you don't come across as condescending or insulting, I think most people would respect your decision and admire your honesty. Many of the consultants would have been in your place a few years ago and they should understand your position. 

P.S - I am speaking from my experience with the firms' India offices, but this should be consistent across national borders. 

Agree with Dolf. I know the person, who turned down McK offer and chose BCG, and then in a few yrs successfully moved to McK.