The emotional bonds behind The Avett Brothers' musical success

From the beginning, confidence was not a problem for The Avett Brothers band, reports "CBS This Morning: Saturday" co-host Anthony Mason.

"We always thought a lot of ourselves early on, more than we should have," said Scott Avett, one of the band's lead singers and founders. "You know, small crowds - we thought they were Madison Square Garden."

"That baseless confidence, it can really come in handy," said Seth Avett, also a lead singer and founder.

The brothers say that it came from their father, who ran a welding crew in North Carolina.

"He used to say, 'No matter what you do, you'll be the best at it.' He said, 'If you collect garbage for the city, there's nothing wrong with that. Just you will be the best at it. Just be the best,'" Scott said.

The band came together in early 2002, when the brothers brought Bob Crawford in on bass after an audition in a parking lot.

"Banjo, guitar and bass in the parking lot. Played about four or five songs," Bob recalled.

Bob would later bring in Joe Kwon in on strings.

"When I first saw them play, that was the first image in my head. I was like, 'My God, they're like Muppets bouncing around on stage,'" Joe said. "You know, it's just an infectious energy that I couldn't sit still."

They built a steady following, breaking out in 2009 with their album, "I and Love and You."

But two years later, Bob had to take a year off when his 22-month-old daughter, Hallie, was diagnosed with a rare pediatric brain tumor. Band members took turns by her bedside.

"There's so many memories of these guys coming from very far away and just... sitting in her room, sitting in the waiting room, outside the pediatric intensive care unit," Bob recalled. "I'll never forget that."

Bob returned with a battle cry written on his guitar that reads, "This machine kills cancer."

"And so by putting it there with the 'I love St. Jude' sticker, where she was treated, kind of allows me to keep it on the outside while I'm trying to keep it on the inside," Bob said. "She's a joyous miracle."

In a letter announcing their new album, "True Sadness," Seth Avett wrote: "Somewhere along the way, the line between music and life faded."

After Seth split with his first wife after a tabloid romance with "Dexter" star Jennifer Carpenter - who he married this year - he, too, wrote about it in "Divorce Separation Blues."

"I was driving up through the snow in... North Carolina and it kind of hit like a bolt and all the verses started flowing through," Seth said. "And it's another question of, 'Is this something we share with the world or not?' And that's a conversation for the majority of the songs that I write."

In this case, Seth said writing about the separation was a "given."

"The Avett Brothers band dynamic with the audience is not -- it's not shallow. You know, like it's not a selfie and then forgotten," Seth said.