Yesterday was like any other day in destiny a bit lag here and there, but today I got kicked out of destiny twice which also got me kicked out of Xbox live. Everything is working perfectly fine around my house but my Xbox one/Destiny. I don't know if it's me or if bungie is currently updating.
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Are you using WiFi? If so, please try switching to a wired connection temporarily. This is an important first troubleshooting step to rule out wireless signal strength/stability as a possible cause. If you're being disconnected from Xbox Live it would suggest a general networking problem.