After Lyanna was kidnapped (or eloped) with Rheager, Lyanna's brother Brandon went to King's Landing to clear the matter where he, along with his father, are killed by Aerys. But why didn’t Robert accompany them? He was betrothed to Lyanna and if he really loved Lyanna, then why didn't he go to King's Landing to justify things? He only rebelled when his life was at stake as Aerys asked for his head from Jon Arryn.
Because things didn't play out as smoothly as that. Brandon found out about Lyanna's kidnapping on his way to Riverrun for his wedding to Catelyn. So instead of continuing to Riverrun he, and some companions rode to King's Landing demanding to see Rhaegar. Aerys had them arrested and then summoned their fathers to court for the trial but instead of the trial had them all killed.
"He was on his way to Riverrun when..." Strange, how telling it still made her throat grow tight, after all these years. "...when he heard about Lyanna, and went to King's Landing instead. It was a rash thing to do." She remembered how her own father had raged when the news had been brought to Riverrun. The gallant fool, was what he called Brandon.
Jaime poured the last half cup of wine. "He rode into the Red Keep with a few companions, shouting for Prince Rhaegar to come out and die. But Rhaegar wasn't there. Aerys sent his guards to arrest them all for plotting his son's murder. The others were lords' sons too, it seems to me."
"Ethan Glover was Brandon's squire," Catelyn said. "He was the only one to survive. The others were Jeffory Mallister, Kyle Royce, and Elbert Arryn, Jon Arryn's nephew and heir." It was queer how she still remembered the names, after so many years. "Aerys accused them of treason and summoned their fathers to court to answer the charge, with the sons as hostages. When they came, he had them murdered without trial. Fathers and sons both."
A Clash of Kings, Catelyn VII
The thing is here Robert was at the Eyrie with Jon Arryn as his ward at that point. Jon is an incredibly smart and collected man and would have advised against Robert and Eddard marching down south. Brandon, was similar to Robert, a hothead, and marched straight down there the second he heard with no one to really council him against it. And that is assuming they had even heard of Lyanna's kidnapping before his brother was arrested and his father sent for. Remember message times are not instant in Westeros.
It's all just a matter of timing and location really.
Brandon Stark was at Riverrun for his own future wife (Catelyn who later married Eddard) and went as soon as he heard. Eddard and Robert didn't hear about the events until Jon Arryn got the command to bring both their heads to the king. Jon refused and started the rebellion with Robert and Eddard at his side.