To start with the game is peer to peer and doesn't use servers for game play. Servers only serve for matchmaking and to hold character data. There are no servers to be sorted out. "Speed" has very little to do with Destiny. Vanilla Destiny only needed about 1Mb up, since the expansions you may need a little more than 1Mb up but it is still a minimal requirement. There are so many more things that go into having a good, stable connection. Latency, the time it takes data to go from point a to point b is much more important. High latency will equal lag and possible disconnects. Latency can be affected by an array of things from equipment/setup problems inside your home network to issues with your ISP. Packet loss is another one. Any packet loss in Destiny will kill your connection. Jitter will also cause problems. Look through the network setup guide and make sure your network is setup correctly. Run tests on your network using the various free tools that are out there to check for jitter, latency, packet loss ect.

The game has been having many connection issues recently. Bungie is working as hard as they can on making the experience the best it can be for all guardians.

Try and check what your NAT type is, this is a know issue