From my experience, it doesn't. All of my kids are fluent in English, Dutch and at least one more language. And I credit early exposure and the schools spending a lot of time on this from age 6 and up.
Our music also contains more English music than German, which doesn't seem to be the case in Germany.
So in the Netherlands, we're exposed to English a lot more than German, French or Spanish speaking countries. I bet the same applies to the Nordic countries, which also have great English speaking ability
What do you believe Dutch schools do well when it comes to learning English? How large are the classes? Are there any things outside the school that might play important role in learning English?
Could it be just the language compatibility? (not sure how Switzerland would fit)