Accents are complicated, but as a general rule, you end up stuck with whatever your brain has learned by age 14. Stephen Pinker has a story about Henry Kissinger and his brother moving to the US just before and after that threshold, one having a German accent and the other doesn't.
Most foreigners never lose their accent, while children of non-native speakers learn the local language without their parents' accent. Natives of Germanic languages, as someone else pointed out, tend to be better at English than natives of Romance languages. And some native Greeks can speak with such a perfect Spanish accent that when their vocabulary lags behind they sound slightly aphasic (until they tell you that they are actually learning the language).
It also depends education: Portugal and Greece used subtitles on tv whereas France, Spain and Germany usually dub foreign films. This makes a difference between the students from those nationalities one encounters, say, in Britain.
I know of a Japanese whose Spanish, learned in a couple of years as an adult, has far less Japanese accent than his English, learned and practiced since primary school.
And now that I'm at it: as with the deaf/mute confusion, when teaching, or just speaking, to a non-native speaker, remember that, rather than having a sticky tongue, they might just not be able to hear the sounds you are saying. This causes much frustration because you repeat the same word ten times, assuming that they can hear the difference, but they cannot. The solution is to point out explicitly in which sound the difference occurs.
Apologies for the dogmatic speech, it's the foreign-sounding generalisation I have a general perception that Europeans are generally excellent at English that triggered it :-)