Here's one answer: No, a story can't have too much dialogue. Look at Dorothy Parker's celebrated short story “Here We Are," which follows a newlywed couple on the train ride to their honeymoon destination. In the short story collection Fiction Gallery, “Here We Are" takes up approximately nine pages. The opening paragraphs are narrative and the rest of the story is almost entirely dialogue. I didn't count how many words make up narrative after those opening paragraphs, but I could have without losing too much time. Plenty of stories, including many by Ernest Hemingway and Raymond Carver, rely heavily on dialogue.
Of course, the opposite answer is true, too: Yes, it's possible to use too much dialogue. If the dialogue is unnecessary, it's “too much." That might happen in a story packed full of dialogue or one that has only a few lines of it. Exchanges should move the story forward and reveal something significant.
Let's look at a scene from Raymond Carver's short story “Whoever Was Using This Bed." Iris and Jack wake up to the phone ringing in the middle of the night. The call is a wrong number, but the alarm of it lingers. Back in bed, their conversation goes like this:
“How about you turning off your light, honey?" I say, as nice as I can.
“Let's have a cigarette first," she says. “Then we'll go to sleep. Get us the cigarettes and the ashtray, why don't you? We'll have a cigarette."
“Let's go to sleep," I say. “Look at what time it is." The clock radio is right there beside the bed. Anyone can see it says three-thirty.
“Come on," Iris says. “I need a cigarette after all that."
I get out of bed for the cigarettes and ashtray.
Each character has an agenda. Jack, the narrator, expresses his desire to go back to sleep and Iris responds by expressing her own desire for a cigarette. Jack tries a second time to convince Iris they should sleep and Iris insists on a cigarette. This could go on and on, but Jack gives in and gets the cigarettes. In this exchange, Carver reveals an important aspect of their relationship. This is essential as the story continues and they discuss deeper, more meaningful matters, like the decision to take a person off life support. Each character comes down on a different side of the issue. Iris wants Jack to “pull the plug" for her if it's ever necessary and Jack wants to be “hooked up just as long as possible." Iris' indifference to Jack's desire to go to sleep doesn't necessarily mean she's going to ignore his wish to stay “hooked up," should that decision come about. But it does work in concert with other moments to show she's dismissive of his thoughts and feelings and that introduces doubt.
Relying on dialogue may be a stylistic choice, but an exchange shouldn't be an indiscriminate volley of words. It should have purpose.