Server timing out, then thinks players are still logged in

Players are getting timed out on our server and being kicked as a result. When they try to reconnect, the server indicates that they are still online. 

They are unable to log in until I go into the server window and click 'enter', at which point I get this output:

[19:48:01] [Server thread/WARN]: Can't keep up! Is the server overloaded? Running 2011ms or 40 ticks behind
[19:49:56] [Server thread/WARN]: Can't keep up! Is the server overloaded? Running 2828ms or 56 ticks behind
[20:03:37] [Server thread/WARN]: Can't keep up! Is the server overloaded? Running 2586ms or 51 ticks behind
[20:06:47] [User Authenticator #25/INFO]: UUID of player Clonedelta is 078f5395-6796-48a3-b0f9-7afe96dcb01d
[20:05:35] [User Authenticator #24/INFO]: UUID of player Clonedelta is 078f5395-6796-48a3-b0f9-7afe96dcb01d
[20:04:57] [User Authenticator #23/INFO]: UUID of player Clonedelta is 078f5395-6796-48a3-b0f9-7afe96dcb01d
[20:04:43] [User Authenticator #22/INFO]: UUID of player Clonedelta is 078f5395-6796-48a3-b0f9-7afe96dcb01d
[20:04:24] [User Authenticator #21/INFO]: UUID of player Clonedelta is 078f5395-6796-48a3-b0f9-7afe96dcb01d
[20:07:33] [Server thread/INFO]: Clonedelta lost connection: Timed out
[20:07:33] [Server thread/INFO]: Clonedelta left the game
[20:07:33] [Server thread/INFO]: com.mojang.authlib.GameProfile@75794c7f[id=078f5395-6796-48a3-b0f9-7afe96dcb01d,name=Clonedelta,properties={textures=[com.mojang.authlib.properties.Property@34506538]},legacy=false] (/68.61.28.238:3048) lost connection: Disconnected
[20:07:33] [Server thread/INFO]: com.mojang.authlib.GameProfile@16799212[id=078f5395-6796-48a3-b0f9-7afe96dcb01d,name=Clonedelta,properties={textures=[com.mojang.authlib.properties.Property@78dcc4e2]},legacy=false] (/68.61.28.238:3056) lost connection: Disconnected
[20:07:33] [Server thread/INFO]: com.mojang.authlib.GameProfile@71bc30d[id=078f5395-6796-48a3-b0f9-7afe96dcb01d,name=Clonedelta,properties={textures=[com.mojang.authlib.properties.Property@22c9c59]},legacy=false] (/68.61.28.238:3064) lost connection: Disconnected
[20:07:33] [Server thread/INFO]: com.mojang.authlib.GameProfile@608cc6f3[id=078f5395-6796-48a3-b0f9-7afe96dcb01d,name=Clonedelta,properties={textures=[com.mojang.authlib.properties.Property@2222a87a]},legacy=false] (/68.61.28.238:3163) lost connection: Disconnected
[20:07:33] [Server thread/INFO]: com.mojang.authlib.GameProfile@5fa96429[id=078f5395-6796-48a3-b0f9-7afe96dcb01d,name=Clonedelta,properties={textures=[com.mojang.authlib.properties.Property@18c95de9]},legacy=false] (/68.61.28.238:3268) lost connection: Disconnected
[20:07:40] [Server thread/WARN]: Can't keep up! Is the server overloaded? Running 228064ms or 4561 ticks behind
[20:08:08] [User Authenticator #26/INFO]: UUID of player Clonedelta is 078f5395-6796-48a3-b0f9-7afe96dcb01d
[20:08:08] [Server thread/INFO]: Clonedelta/68.61.28.238:3292 logged in with entity id 753856 at (-947.5, 10.0, 203.3878258892246)
[20:08:08] [Server thread/INFO]: Clonedelta joined the game
[20:08:48] [Server thread/WARN]: Can't keep up! Is the server overloaded? Running 2218ms or 44 ticks behind