It depends what you mean by “sound like a native speaker”; lots of non-native English speakers here write much better than many native speakers. Some even speak better, in my opinion. Chinese and English are entirely unrelated, with radically different phonology, lexicon, grammar, and idiomatic expressions, so you should be congratulated on achieving any level of communication whatsoever.

Unfortunately, not every English speaker is aware of that (truth be told, most are not). A foreign accent can sound pleasant, but if your pronunciation is inaccurate, it will be hard for people to understand you. Other people from your own country will understand you (or you could just talk to each other in your own language…), but natives will have trouble and so will non-natives with a different foreign accent. But not being understood is the least of your troubles — the way you speak is the way you expect to hear, so if your pronunciation is inaccurate, it’s going to be a lot harder for you to understand the spoken language than it needs to be. Add up inaccurate pronunciation, grammar, word choice, and so on, and you can forget about communication — it’s not going to happen. By the way, I think “finger out” is cute, but it might be better to stick to the usual expression (with “figure”, not “finger”). Let's just keep in mind that learning a language takes time, and if you aim for the stars, then even if you fall short, you’ll certainly go as far as you can. And that’s farther than you may think.

When people say they wish to "sound like a native speaker", they mean that they want their pronunciation to be as close as possible to that of native speakers. It does matter very much if your purpose is indeed communication. The further one sounds from a native speaker, the more likely it is that people will switch to English, underestimate your ability to understand and speak the language, and the less likely it is that you will be accepted as one of them. I don't like it when I'm at the stage I'm told I have a cute accent or way of pronouncing words in the target language. It just reminds me that I still have a lot of work to do.

The other point I will make is that language is not just for communication. Each language has its own norms and character. It's a sign of respect to make an effort to learn the correct pronunciation and speak in a way that native speakers don't have to struggle to understand you. If you merely transfer the sounds of your native language and speak any old way, it can come across as rude and it seems like you're placing the burden on them to decipher your speech. When you try, you're showing that you're taking the initiative to fit into their community and taking responsibility.

And sorry for chiming in again; I didn't quite figure out what I actually found odd about your question yesterday.

And now, as I reread the thread and some of the other responses, it suddenly struck me that maybe how you think of communication profoundly differs from how I interpret the word.

To me communication is a two way process, and both the speaker and the listener are involved. It's about expressing and understanding. Maybe to you (given the emphasis on "cute") it means the same as speaking without caring whether you're understood?

Well said Phil. Because of your aim that determines your level's pace.

For me. I live in an English speaking country. But my aim is to earn enough money to support my family. So whatever degree or successful life is not a matter of me. I don't need to learn English like a native level or native accent. I just using understandable and flaw English is ok. I just am communicable and let others understand that is fine. But if for a teenager or who immerse in an English environment. You need to speak like a native or better English you can. Because it is better opportunity for them. And avoid others look down your abilities. Someone is advanced learners that know a lot of profound words and natural ways of speaking but they have little accents that I have came cross often. That is apparently their flaw that showing that he wasn't a natives but he was able to be a college teacher. He was my previous community college teacher. It is being said that you have accent who still can be successful one. But it labels you are not a local or native. Someone will judge that by your inaccurate word choice and accent . If for little kids there is few play mates with in the school. So you will feel the down side. Back to me. I don't need to be like a native but I reply in English because most ones can read it and if I reply in Chinese that means burning the English speakers in earth. They don't get what I deliver. So language is just a tool to communicate but whatever level you achieve you just enjoy it. No matter other judges you but accents are not your fault. If you have flawless accent you are incredible impression of other. You are young you definitely can remove the accent.

Just a comment from my personal experience.

If you speak a foreign language you should try as hard as you can to sound like the target audience (the natives). I live in an English speaking country, there are people from all over the world here (including myself), and while I'll never shake my accent (well, I can, if I try hard, put on a local accent and get away with it, but that takes the fun out of communicating because I have to concentrate too much) I am usually well understood by the locals (the majority thinks I'm a native English speaker with a South African accent).

Some of my colleagues (past and present) have both a mixed level of English in terms of correctness, but also in pronunciation. And while someone with a strong Indian accent can sound pleasant (as in the listening experience) I find them next to impossible to understand, because they don't have the same rhythm and pitch, and run their words together. Sure, that can sound cute, but if your colleagues have to keep asking you to repeat what you said two or three times it gets old; and it clearly defeats the purpose of "communication".