Both servers were mismanaged. BE not doing well was mostly Tok's, Hanman's, and Zirak's fault. All three were pretty rude to most players. Tok and Zirak would either rarely update the game in a positive way or would make shit updates that made the game worse (at least by most people's opinions). For example, Zirak made a patch to the game that prevented a large amount of the playerbase from actually logging on to the game in 2012, which resulted in the playerbase being drastically decreased. Before that update, there would usually be ~40 players or so online at any time, but after that update there would only be 2-3 people. In addition, Zirak did something really retarded with BotStract. He made it so that having more BotStract would make your bot jump higher to the point of it becoming completely ridiculous. He also changed the "traditional" +5 system to a +9 system that most of the playerbase didn't want. Those things amongst many others resulted in BE dying.
I'm not really as familiar with BC since I wasn't here at "new" BC since the very beginning. I came here after BE started to die. I think that BC died largely because people just got tired of the game. Most of the updates were just reskins of assets that already existed that had way broken stats. Basically, that means that there were no real updates of substance if you weren't into mindless grinding sector to get GP. I do think BC did a lot of things right though, namely largely improving on BE's [C] tag, offering a lot more gameplay options in PvP, and reducing chain hits. That still doesn't change the fact that there was no real brand new content in PvP or BvB though. Most of the people here who browsed the forums that have been playing Bots off and on for several years just got tired of grinding sector mode, so I think that the demise of BC was kind of inevitable.
Also, T-Bot was shit because they did janky shit with the running speed of the bots. All the bots were really slow, so much so that running+jumping was faster than just running. I'm pretty sure that Fira and Kenny DDOS'd the server frequently as well.