I agree with martin on the usage of the word meeting. It doesn't sound native. As for your sentence, a native would say something like this, "We didn't win the meet, but we did finish third" - If you are speaking positively or joyfully about the result "We didn't finish first in the meet, but we did get third place!" - If you are speaking positively or joyfully about the result "We didn't win the meet. We got third. - If you are speaking negatively or are sad about the result "We didn't get first. We finished in third place." - If you are speaking negatively or are sad about the result

Your sentence makes sense but, its just not something you hear often in the US. Your probably more likely to hear the actual event name(soccer tournament, tennis match) or maybe just the word game. If you said sports meeting in the U.S. people would likely think you were talking about a discussion about sports. "Meeting" is more often used to refer to a group discussion. Native speakers from the UK may not agree with this but, If you are going to use meeting, cut off the ing and make it a noun, "sports meet" is grammatically correct as it describes an event not an action. What you say is always greatly determined by your target audience.