of course. the majority of us are not able to, but many babies do it everyday. it is very difficult to do it own your own, but certainly not impossible. however, the majority of speaking language correctly is the subtle nuances that come with speaking a language. things like what emphasis to put on what syllable in a word or how to roll your "r"s. these are things that cannot be taught.
the english language has a lot of slang when we speak. other languages have this same thing that you wont get from teaching yourself a new language.
the answer is yes you can. though, professional instruction will give you many things you may have missed out on. self instruction like the Rosetta Stone is a great way to teach yourself a new language.