SpongeBob wakes up, stubs his toe, and starts crying. He rips his favorite shirt and starts crying again. He goes downstairs and hears a sad song on the radio; he cries again.
At work, he flips a Krabby Patty, which lands on the ground when Mr. Krabs startles him. SpongeBob then cries once again. Squidward gets irritated of SpongeBob's crying, and he yells at him. SpongeBob then cries, wailing, "You yelled at me!"
Squidward tells SpongeBob that he cries too much; SpongeBob denies it. Squidward then pulls up a TV and plays a video of a trillion tears and millions of places where SpongeBob has cried over those years. He then makes a bet with SpongeBob that he couldn't last until midnight without crying.
After the bet is made, Squidward follows SpongeBob home to make sure he cries, and tries to make SpongeBob cry over things like a beautiful sunset, sad news stories with funerals, a sad song on the clarinet, and a sad bedtime story.
Amazingly, SpongeBob manages to hold in his tears until midnight, despite all of Squidward's best attempts, and after midnight, SpongeBob cries all over Bikini Bottom like crazy. Squidward comes over for a slumber party the next night, and the episode ends with Squidward crying and SpongeBob consoling him.
Patrick is correct after he answers SpongeBob's question: it's possible for somebody to not try cry/sob for at least a day or two.
However, because of SpongeBob's strong sensitivity over something dramatic/emotional, it is impossible for him to overcome a day without crying.
This episode is similar in concept to Funny Pants as in both episodes Squidward gets annoyed by SpongeBob's emotions (in this case, Squidward gets annoyed at SpongeBob's excessive crying while in Funny Pants Squidward gets annoyed at SpongeBob's excessive laughter).
On the radio, the announcer said the song was by Eely Dan. This is an obvious reference to the jazz rock band Steely Dan. Also, the next song was by Lionel Fishy. This is also a reference to Lionel Richie.
The radio announcer also states that the song's name before "Sad Songs" was Aja, a song by Steely Dan on the album of the same name and a song called "Sad Song" by Paul Williams.
In the subtitles on TV, SpongeBob says "Stop. It burns," when he would normally say "Sob. Whimper."
SpongeBob can be seen shedding a tear during the news story about the snail, which means Squidward won the bet all along.
When SpongeBob cries at Sad Song, his tears come out his upstairs bedroom's window, but he was in the kitchen, completely opposite, and downstairs, from his bedroom.
During the slideshow Squidward made of SpongeBob's breakdowns over the years, SpongeBob wore his Krusty Krab hat in all of the clips even though most weren't set at the Krusty Krab.