Weasel is an error code that indicates general networking issues - your network is having difficulty maintaining a connection to the Bungie servers.
I highly recommend taking some time to go through the Network Troubleshooting guide, and apply steps within:
https://www.bungie.net/en/Support/Troubleshoot?oid=12915
Destiny certainly requires a strong and stable network connection, more so than even many other games, due to the nature of the game. The ability to connect to other games, does not mean that there aren't steps to be taken on your end that may improve your connection to Destiny.
Give the guide a try!
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Same problem here. It's not viable to go alone on heroic assaults or others high light level events, cause i get disconnected via weasel too frequently, most of times when i'm near defeating a boss or heavy enemy waves... TOO ANNOYNG. If I recall, that was why i quit playing Destiny just after Dark Below went out. I spend a lot of time trying to reconnect, reseting, checking NAT status and browsing this foruns, instead of playing and having fun...
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Once again all you have to say is that i need to do things to resolve your problems !! My connection is very stable and have absolutely no problems with any other game with alot more players connected why cant you accept you have a problem and need to sort it