Do you recognize the distinguished looking gentleman in the photo? He was trilingual in Polish, French and English; he wrote outstanding prose in his third
language, English, and he is now recognized as one of English literature's great authors. His name is Joseph Conrad and, as it happens, he had a very strong Polish accent in English!
There is a longstanding myth that real bilinguals have no accent in their different languages. Joseph Conrad and many other bilinguals, in all domains of life, show how unfounded this myth is. Having a "foreign" accent in one or more languages is, in fact, the norm for bilinguals; not having one is the exception. There is no relationship between one's knowledge of a language and whether one has an accent in it.
Researchers do not agree on an accent age limit - no accent if a language is acquired below it, the presence of an accent if it is acquired later. Some have proposed that a language can be "accentless" (in the sense of not being influenced by one's first language) if acquired before age six; others extend the window to age twelve. Personally, I have met bilinguals who acquired their second or third language even later who do not have an accent in it.
Usually a first language will influence a second language that is acquired later, but it is not uncommon that a second language may influence the first. This happens when the second language is used much more than the first, over an extended period of time. Bilinguals who start having an accent in their first language are usually very conscious of it and often comment on it; some even excuse themselves. But it is a normal linguistic phenomenon explained by the circumstances of life.
As for the origin of accents in a third or a fourth language, one must examine the bilingual's language history. It really depends on when and where the person acquired the language and which other language was dominant at the time.
Some bilinguals even have an accent in all of their languages. This happens when they spent their early years moving between language communities. Once again, this is not an indication of how well they master their languages.
We are all conscious of our accents; some see disadvantages to them whereas others see advantages. Among the disadvantages, the one that is mentioned the most is that it makes you stand out from others when you want to blend in. If the society you live in is not positively inclined toward the group you belong to, an accent can have a negative effect on the way you are perceived and treated.
Another disadvantage is that an accent may signal that the speaker has not tried hard enough to learn the language when, in fact, it is due to neuromuscular factors that are difficult to control. Finally, although having an accent does not normally impede communication, when it is very strong, it may do so, even though the person may be fluent in the language being spoken.
But there are also many advantages to having an accent. Some accents are seen positively by people or groups (e.g. a French accent in Germany, a British accent in France, etc.). I have also known of cases in which an accent was a major factor in a person's falling in love with someone (although not the only factor, one hopes).
An accent also clearly marks you as a member of your group if you do not want to be seen as a member of the other group. Linked to this is that an accent can be self-protective. It prevents members of the group you are interacting with from expecting you to know their language perfectly as well as all their cultural and social rules.
In sum, having an accent when you know and use two or more languages is a fact of life; it is something you get used to, as do the others you interact with.
Note: In another post, I discuss the factors that explain why some bilinguals retain an accent while others do not (see here).
Reference: Grosjean, François. "Having an accent in a language". Chapter 7 of Grosjean, François (2010). Bilingual: Life and Reality. Cambridge, Mass: Harvard University Press.
"Life as a bilingual" posts by content area: http://www.francoisgrosjean.ch/blog_en.html
François Grosjean's website: www.francoisgrosjean.ch