It seems to me one disadvantage to learning a new language via a traditional grammar approach would be time. If a student begins studying a language while in school, concurrent with English (or whatever primary language) grammar, this approach is obviously preferred. If not, if one simply needs to be able to function quickly in another language, there are better, more expedient approaches.
The traditional approach of studying grammar is too isolated to really help someone learn a language. There is no real context or application given when learning the parts of speech, sentence part, clauses and sentence types. They must be given opportunities to learn the structure of written language and then an opportunity to practice this in meaningful written and spoken ways.