One of the disadvantages of bilingualism is that you might mix up your languages when you speak. How so ?
If you speak only one language, your reference points are limited to what is correct within that particular language structure. Your brain encodes this; everything outside this perimeter is registered as "alien" or "wrong." When you introduce a second language, your perimeter widens to accept new rules and applications as being correct as well. If the two languages concerned are very different from one another (for example, Japonese and Spanish), there is usually no mixup problem since the difference in signals sent is very clear.
However, if the languages are similar, confusion can happen since your mind can momentarily encode one language structure as being part of another. (Similar to jumping from one soundtrack to another by mistake or having simultaneous radio reception from two different channels).
The chance of this happening is greater if the speaker is continually jumping from one language to another instead of staying within a "pure" linguistic environment. A common case is the bilingual family. Ideally, to avoid confusion each parent should only speak in his or her native tongue, at least when addressing the children. In that way, the languages are encoded separately and (to the child) the distinction between the two is clear.
For the same reason, it is not a good idea to take on two new languages to learn at the same time. The perimeters of one language need to be clearly established before introducing another.
Below are references citing two particular case studies.
It is interesting that you see this as a disadvantage. Perhaps the use of words from two languages in oral conversation can be considered 'speaking bilingually'. There are words and phrases in different languages that can be used to describe a concept with more precision, and so the use of words or phrases from a second language often adds to a conversation. The English speaking population in Quebec have been studying in French Immersion for more thatn 30 years, and do very well on standardized tests. Montreal also has the largest trilingual population in Canada and again this is seen as a positive thing. I suggest you look at second language research, particularly by Fred Genessee at McGill University. In conclusion from his paper 'What do we know about bilingual education for majority language students' he states "bilingual education for majority language students is effective in promoting functional proficiency in a second, or even third, language at no cost to the participating students native language development or academic achievement".