Speaking English like a native speaker is very good for anyone who is going to deal with people who are actual native speakers of the language. Doing business with someone who really speaks your language well makes you relax and feel better about that person. It is also very impressive.
The only drawback I can see comes when you deal with people who are not native speakers -- particularly from your own country. If they all speak English in some "improper" way and you speak it "correctly," you will stand out and you might be accused of not really being like other people from your country.
I think it's both a strong point and a weak point. but to some extent, what do you mean by native speaker? Who exactly? Singaporian, American, British or Australian...? The language i wish to speak like a native speaker is American English (General American English=standard dialect=neutral English)
Strong point if the language you speak is widely understood by other native and non-native speakers. In this view, you have to speak what is a so-called "neutral English" as i mentioned.
Weak point is that the language you using even like native english speaker but it's hard for other to understand you even native English speaker. Obviously, in a native English country, we've already known, there are many regions with different dialects which others can not understand with their great efforts.
to sum up, speaking English like a native English speaker doesnot always mean "good". It depends on someone' s view point. In my oipion, whatever kind of English we use, we have to consider "who is our audience?" to choose the right accent to get their attention and heart. Our accent is not always stationary. we have to adapt in order to get success in our communication!