Say a game company goes bankrupt but another company wants to make a sequel. Is it possible for the other company to obtain rights to make a sequel?
Who would they even contact about this?
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A company does not cease to exist simply because it goes bankrupt. The company may wind down its operations, but it may just go through a process of restructuring its debts.
If the company is merely restructuring, the bankruptcy would not probably not have much effect on its ownership of intellectual property such as copyright and trademark rights. So it would retain those interests in the video game, and the company making the sequel would need to obtain licenses from the original company.
If the original company is going through bankruptcy in anticipation of shutting down completely, you can expect it to sell off all its assets, which would include its copyright and trademark interests in the game. In that case, the company making the sequel could either attempt to purchase those rights itself, or it would need to obtain a license from whoever does buy those rights.
The copyrights do not disappear just because the owner ceased to exist. That applies to both companies and people. As an asset, the copyright is part of the estate of the entity and will be either transmitted to the heirs, remain in the estate, or go to the buyer of the assets. As @MSalters pointed out, assets that are not sold will belong to the owners of the company.
It might become nebulous as to who is the owner in the case of a company going bankrupt, but the problem arises when someone who has a plausible cause to believe they are the owners decides to sue.
The game company will be dragged into a lawsuit that might end either way. Sometimes the lawsuit in itself is costly enough (in both time, effort, and/or money) to discourage even touching that IP.
One real-world example was Sunbow Entertainment  that went bankrupt and some of its assets went to Sony Wonder and some to Rhino Entertainment. One IP in particular, Jem and the Holograms, rested in copyright limbo for years until Hasbro bought the rights to both sides.
If someone were to make a product based on Jem and the Holograms during that time, a lawsuit could have come from Sony, Rhino, or even Hasbro, as any of them could have claimed to be the rightful copyright owner.
Only after Hasbro cemented its claim did the company decided to try and flop at making a live-action movie based on that IP.
Licensing would be a nightmare, as every one of the possible copyright owners would want their share of the pie.