There is no hardwired 'limit' on the number of languages a child can learn. In fact, a scenario where a child learns one language for each parent, a language of the community they live in, and a language used in school, is not that uncommon.
However, you must understand that learning a language is not an on/off proposition. People learn different languages in different contexts, for different purposes. So you cannot simply expect somebody who simply grows up will be able to become a professional translator or even use all languages in all contexts with the same facility. This will be particularly obvious for literacy contexts. For example, unless you give them practice and some instructions, your children may struggle with writing in reading in the languages that they only know from spoken contexts. It may sound obvious but many people assume that writing is just writing speech down. It's much more.
You also need to be prepared that there will be a developmental time during which the child will appear to mix the languages and this mixing will to some extent continue as code switching into adulthood. There's nothing wrong with that but you might get negative feedback from people who only experience monolingual contexts.
Another thing that often happens is that children themselves end up resisting using some of their languages because they don't want to stand out from others. Some children are reluctant to speak back to their parents in the language of the parent (even though they understand) if that is the only context they ever hear the language (and if the parent speaks the language of the community in which they live).
But the key thing to remember is that there's no harm to learning multiple languages from early childhood. There are also some benefits to multilingualism (although I suspect that people overgeneralize from the available research).
On the other hand, it's also not uncommon for children not to be able speak the native language of one or both of their parents. There are people who cannot even pronounce the name of one of their parents (or even their own) in the 'native pronunciation'. While many people decry it as a missed opportunity, it's important to remember that this is as natural as multilingualism and need not be stigmatized any more than teaching children multiple languages used to be.