Often you can and in those cases Japan generally requires you to. However it would depend on the citizenship in question.
My understanding (again, not a lawyer) it that this could have legal implications if you do something of your Japanese citizenship after reclaiming a foreign citizenship. Although apart from running and winning an election, or maybe holding a job restricted to Japanese citizens, I'm not quite sure exactly how you could get in trouble.
The loophole dual citizens (since birth) use is actually just that: since they never acquired a foreign citizenship (they are born with it), they never lost the Japanese one... so all Japan can do is order them to make unspecified efforts to renounce their foreign citizenship. Looking it up on the internet and concluding it's complicated seems to count as "efforts".
And if you lose your Japanese citizenship, you go back to being a foreigner. Without a visa. Meaning you risk losing your ability to live and work in Japan.
As most people who naturalize do so because they've established deep for-life roots in Japan, their Japanese citizenship is not something they're willing to gamble.
For someone who gained JP citizenship through naturalization, it's a different matter. From what I've read, they will check to make sure you renounced your old citizenship, and will revoke your new JP citizenship if you don't (which probably also means no visa eligibility).
From what I've heard it goes like this: British person applies for US citizenship. He has to "renounce" his British citizenship and hand over his British passport to the US authorities, who shred it. The next day he goes to the British embassy, tells them what has happened and asks for a new passport. The British authorities give him a new passport and, of course, they don't say anything to the US authorities.
(There might be different cases to consider here, depending on how you acquired your British citizenship. As I understand it, if you were born in the UK of British parents then you have a right to a British passport regardless of how old you are, where you have lived, and what other passports you've had. It's not possible to "renounce" that right. It might be different for someone who acquired British citizenship in some other way: perhaps some people could "renounce" their citizenship.)
>U.S. law does not mention dual nationality or require a person to choose one nationality or another. A U.S. citizen may naturalize in a foreign state without any risk to his or her U.S. citizenship
https://travel.state.gov/content/travel/en/legal/travel-lega...