The main advantage of learning two languages at once is that you are more likely to maximize how quickly you can learn both languages. In other words, let us say that it would take you 2 years to learn a language if you studied that language exclusively. It then stands to reason that it would take 4 years to learn two separate languages consecutively. Now imagine that you are learning two languages at the same time. You might be able to learn both of them in 3 years rather than 4. It is at least possible that you will devote more time to language learning overall if you study 2 languages at the same time. You might be more willing to spend, for example, 1 hour per day on each language than you would be to spend 2 hours per day on one language.
Some people also think that learning two languages at once makes it easier to learn. In a sense, your brain is being challenged more because it is learning two languages and so it becomes more receptive to learning a language in general. In addition, languages that are similar or that have many words that are similar may be easier to learn at the same time.
However, many people also think that it is hard to learn more than one language at once. The main disadvantage of doing this is that there is a relatively great danger of confusing the languages that you are learning. In this point of view, if you are trying to learn two languages at once, their vocabulary and grammar will get mixed together in your mind. Conversely, if you truly master a language, it becomes a language that you know well and not one that you are learning. Your brain does not pull up words from that language when you are trying to learn a new language. For example, I personally speak English and Language A fluently. I am conversant in Languages B and C and I am trying to learn Language D. When I try to speak Language D, I never put in English words or words from Language A because those are languages I truly know. My brain will, however, sometimes put words from Language B or C in when I try to speak Language D because all of those are (as I think of it) in my brain under the category of languages I am learning.
I believe that the advantages and disadvantages of learning more than one language at a time have a lot to do with your own personality and your own aptitudes. I am not sure that there is a “one size fits all” answer to whether it is a good idea to study more than one language at once.
See the link below for a discussion thread on this topic form a language-learning site.
I once took introductory French and introductory Spanish at the same time and even had one class right after the other one. I got A's in both--but of course they were only introductory. I see no great advantage in studying two languages at the same time. The disadvantage is that you have to memorize twice as many foreign words, conjugations, rules, etc. One proof that it is disadvantageous to study two foreign languages at the same time is the fact that few people do it. Most of us are crazy enough without adding to our confusion that way! I should think it would be OK to study two languages after one had a good knowledge of both languages. That would get you into comparative literature, which could be interesting and could even lead to a Ph.D. in Comparative Literature. But I don't think it is wise to try to learn two languages at the beginning level at the same time. Doing so might prevent you from focusing on either language. I think the best way to learn a foreign language is to saturate yourself in it by listening to the language on tapes, watching movies in that language, taking labs in that language, doing extracurricular work in that language, etc. Then if a student is advanced in one language, it wouldn't seem harmful to start learning another language at an introductory level. If a person wanted to study two foreign languages at the same time, it would seem like a good idea to study related languages such as French and Spanish. Of course, you have to keep up with your own language too. That's three languages.
The major advantage of studying two languages at once is the student finally gets insights into what we mean by "language" -- what the ingredients, the mechanisms, of all languages are. When we learn, for example, the tenses, or the numbers, or the declensions, or the modifiers of two languages, we gradually understand what language itself is, what subtleties and distinctions are available for human expression. A student of, for example, French and German can comprehend a question -- "Was bedeutet das wort ---?" and realize that the present tense of the verb is required, not to obey some abstract rule, but to signify that the speaker wants to know what the word means "now," not its original meaning. When we all learned rudimentary Latin in high school, it was the first step in this process, and all the rest of our life we understood English better, because we parsed sentences and wrote lists of verb forms, etc. -- we began to understand structure in language expression.