To make a serious play out of such home-style ingredients is tricky today; a writer risks literal kitchen-sinkiness. Ms. Kennedy seems to have understood this and tried to compensate by hitching her story to a series of symbols — what we used to call in lit-crit classes objective correlatives — that are meant to act as emotional amplifiers and imply a more consequential theme.

In “Napoli, Brooklyn” these correlatives, which the author juggles like oranges, include Tina’s cramped fist, the butcher’s dream of a white coat and Ludovica’s inability to cry. The last has become so severe (and so contrived) that even carrying a cut onion around doesn’t help.

Then there’s the biggest correlative of all, a disaster so central to Ms. Kennedy’s concept that I don’t dare spoil it. (Still, it will be evident to many New Yorkers just from the play’s setting.) As staged by the director Gordon Edelstein, this disaster is shocking and brings with it, at least temporarily, the kind of frisson that can sometimes occur when fictional characters meet real-world events. The problem is that, as objective correlatives go, it’s so overscale it doesn’t really correlate. It overwhelms what it is meant to amplify.

Almost everything that happens afterward feels even falser than the onion. Nic, having experienced what he calls a miracle, is a changed man, dressing up for dinner and crooning with Mario Lanza on the radio. Rapprochements are attempted. Vita returns from the convent at Christmas to find her parents dancing. But none of these changes have been justified in terms of character, only calamity. The story thus ends up shrinking, unable to meet its surreal central image halfway.

What story could? Among contemporary playwrights only a few, at the level of Caryl Churchill and Suzan-Lori Parks and Tony Kushner, can work that particular magic. Still, it’s good that Ms. Kennedy is aiming so high. Also good is the faith the Roundabout has shown in her by commissioning “Napoli, Brooklyn” even before “Too Much, Too Much, Too Many,” her first professional outing, was produced at its Underground stage in 2013. That is how you grow playwrights.

If there are stumbles along the way, there are also many felicities. Sharp Vita begins to appreciate the convent when she notices that a veil she’s nicked acts “like a goddamn spotlight for my cheekbones.” Shut-down Tina (Lilli Kay) starts to unclamp in the quiet company of Celia (Shirine Babb). Mr. Duffy (Eric Lochtefeld) gets a charming series of flirtatious scenes with Ludovica, and a very moving if surprising one with Celia. And when Francesca (Jordyn DiNatale) and Connie (Juliet Brett) — who turns out to be Mr. Duffy’s daughter — invent a substitute for the sex they are not yet ready for, it’s flat-out beautiful.

The acting is terrific across the board, but you can probably sense the way Ms. Kennedy has overentangled her characters, like an anxious sailor making redundant knots. Great plays are freer than that. They aren’t onions trying to make you cry. They are more like ships heading across the ocean, with us as the stowaways.