Yes, and for a learner, it is NOT always possible to hear everything, even though you think you are.
About ten years into my Chinese experience (and having lived in Taiwan for awhile as well) I was in a taxi with another foreign friend (one who was uber-fluent, truly uber-fluent, in Mandarin.) I said something to the cab driver; he repeated something back to me. I didn’t realize that what he had repeated back was actually a correction of what I’d said to him (wrong). My friend asked me just then, “Didn’t you hear him correct you?” And the sad thing was, I had not heard. It wasn’t that I didn’t want to hear, or didn’t care – I simply was not at the point of acquisition (that is, I hadn’t solidly enough acquired the elements that are “lower down” on the structure of the language overall, like the foundation of the pyramid, if you will) to be able to pay attention to the finer point he was demonstrating correctly.
And that makes sense, given the idea that language acquisition (and thus greater accuracy) comes from repeated exposure to meaningful, correct forms (aka “comprehensible input”). That repetition in the correct form was one more toward reaching the critical mass I needed to acquire that particular feature (can’t remember what it was at the moment). But at the same time, for performance in a language at any moment, I like to think about a theoretical model applied to interpreting, which says (basically) that your brain is only so big, and if there are more things you need to pay attention to than you have processing capacity, something is going to get ignored. That’s what happened to me (and still does).
I also did an Oral Proficiency Interviewer training session for Mandarin in which we rated a bunch of speech samples from students of Mandarin. There were 2 or 3 non-native speaking teachers and a bunch of native speaking teachers in the session. We heard one sample from some one who had no tones whatsoever – perfect grammar, easy to understand, but no tones at all. The non-natives wanted to fail him, but the natives weren’t bothered by it at all. “We can understand him just fine,” they said.
I’m not saying ignore tones – far from it. But I think it would help everyone learning Chinese if they were put into their proper perspective. They are important, but not a deal-breaker in all situations, as is often claimed. If everyone could just relax about them, and think that they are (truly) no more difficult than learning to roll your “r” for Spanish – another “important” sound that can be done without and which is actually absent from certain dialects of Spanish anyway, sort of like there is major tone variation in different Mandarin dialects in real life – maybe more people would get to be proficient in Mandarin, rather than giving up after a few afternoons of listen-and-repeat.
There are also two kinds of tone errors – knowledge and performance. The reason foreigner tone errors seem to impact comprehension more might be that these two errors are mixed in together in their speech. A person who has a foreign accent substitutes one sound for another on a consistent basis, so the listener can figure out the “code” pretty quickly. But a person with knowledge errors in tones just plain doesn’t KNOW which tone it should be in the first place, so there’s an element of randomization thrown in. Since tones are just like single sounds in English (think “bat” vs “pat” vs “mat” vs “fat”) – it’s easier to get used to a speaker who consistently says “pat” when he means “bat”, but not so easy to get used to one who more or less says one of these four at random when he means “bat”.
That being said, I firmly believe that most of the grammar issues come from rules-and-output teaching. Students are rushed through the grammar rules, and usually asked to “Zao Ju” (“make sentences”). That means early output and way less input than is needed to acquire the structures. Structures take a long time to acquire – it doesn’t happen in a week just in time for the unit test. The students don’t really understand why they make errors in doing that sort of homework, although they are “corrected”. And anyway, applying rules, even if you know all of them, is time-consuming and leads to hesitation in speaking. The fluent speakers are the ones who can go from the meaning to the Chinese output, without stopping to go through English or use rules to “translate” what they mean into Chinese.