If it is a connection to within the US48, most AA fares only allow 4 hour connection (which is fairly standard among legacy carriers) before stopover occurs and you can only book on separate fares for overnight connections. However, certain smaller, less well-served airports in US48 may allow up to 18 hours (it will be specified in stopovers section of fare rules). See example below from ERI fare rules. Hawaii, Alaska, Canada fares currently also allow up to 18 hour connections. Unless other specified in fare stopovers section, other international destinations allow up to 24 hours.

Many posters here are missing the point. You can always price out overnight connections (with multi-city) and certain websites (like ITA or Google Flights) may offer them even if you don't use multi-city. But they will not always be priced out on a single fare (just because they are on one ticket does not mean there is only one fare involved). If layover exceeds stopover allowance it will be pricing out on separate fare components (you can look at fare rules to determine whether or not this is the case -- if only one fare rule is listed, it's pricing on a single fare).