This is a very difficult question and one that is sometimes hotly debated among both non-linguists and linguists. While I don't expect to end the debate, I can shed a little light on the question that supports the assertion that at times, communication cannot be achieved if there are language errors. Linguist William Labov of the University of Pennsylvania pointed out that on a daily basis, we are misunderstood more often than is commonly believed because of phonetic and other language errors. Additionally, a recent language study revealed that upwards of 60 percent of native American English speakers fail to understand conditional sentences and other higher order sentences, like subordinated sentences, on a regular basis.
One of the illustrations Labov uses to prove language error misunderstandings involves the common officeplace words "bus" and "boss." He finds that surprisingly often these words are misunderstood, thus sentence intent is misunderstood; thus communication is not achieved. In his example, relevant to The Atlas of North American English, he says on NPR [playing a computer recording]:
COMPUTER VOICE: Bosses...
Mr. LABOV: You play that word and everybody talks about bosses, and then they hear...
COMPUTER VOICE: The bosses with the antennas.
Mr. LABOV: And they're wondering who are these bosses with the antennas? And then they hear...
COMPUTER VOICE: I can remember, vaguely, when we had the busses with the antennas on the top.
Mr. LABOV: And they realized that this person is saying the word bus the way they say boss. ... they think they understand everything they hear, but there will be a lot of misunderstandings. (National Public Radio, All Things Considered. Robert Siegel. February 16, 2006)
When Labov conducted his seminal study of African American Vernacular English (AAVE), he operated on the assumption that he could correctly understand the speakers of AAVE: he proceeded on the belief that his understanding of what was meant by the speakers was indeed what they meant. Thus he drew conclusions about the logicality inherent within the construct of AAVE, answering, in effect, "Are these speakers expressing logical thought?" Other study has subsequently been performed that indicates that Labov's team in fact did not understand the speakers and that therefore his conclusions were in error. Thus, Labov may unwittingly have proven his own point that communication cannot be achieved if there are perceived or actual language errors (dialects, slang, creoles have traditionally been described as containing language errors, though they are no longer thus described).
As a result of these studies, which all point to errors in understanding and communication failure, it is quite reasonable and logical to answer the question "Can communication be achieved if there are language errors?" with the conclusion that "No. Communication cannot [uniformly and unvaryingly] be achieved if there are language errors."
The Atlas of North American English [Entry page]
Yes, communication can be achieved even if there are language errors. Simple errors can be corrected or understanding checked by asking what the other person understood. If the error is in a different language, the checking must be done very carefully as even the checking questions could be error-filled. Errors make it harder and could even be dangerous if the incorrect communication is, for example, an interpretation of a medical question or answer with errors. We all need to be more conscious of errors and miscommunication as it happens so easily and can lead to serious consequences.