I spent two weeks in Berlin (in the then East Germany) as a guest of a family of a young Berliner I befriended a year earlier.
This was in 1968. I was 18. My friend mentioned JFK at least once a day, so great was the impression he had made.
The reason why my 17-year-old Berlin friend (and all the other Berlin teen-agers I met) was so impressed, however, stems from accents, believe it or not.
In the Berlin dialect of German ich (I) is icke, when used by itself, and ick when used within a sentence.
What Kennedy really said was not "Ich bin ein Berliner" but "Ick bin ein Berliner." The people of Berlin (well, at least the teen-agers) loved him for not just saying it, but saying it using the Berlin dialect. Because Kennedy used ick, he was instantly accepted as a genuine Berliner.
Now, more than 30 years later, I realize that Kennedy was actually lucky: He most likely was not aware of the local dialect and meant to say ich, but as a typical American was unable to pronounce the guttural ch sound, and said ick without even knowing he was actually making a mistake. The fact that American sources always quote him as using proper German ich, rather than the Berlin ick is the main reason I now believe it was just a lucky mistake.
I am not writing this to suggest that the Berliners would have loved Kennedy any less had he said ich, just that the impression he made was intensified by his unwitting use of Berlinese.