I applied through an employee referral and heard back within one week of sending in my cover letter and resume in. The first part is a quick 20 minute screening to hear about your background, previous sales experience, and why you want to be in the program. The second was a 30 minute Skype interview with a hiring manager with straightforward interview questions. The hiring manager was very conversational and was just trying to gauge who I was and why I wanted to work at LinkedIn. I mentioned the values, mission, and vision and why they were important to me. They are truly looking for someone who will be a great cultural fit.
I found out that day that they would fly me out to San Francisco to interview and meet current members of the BLP program. There were about fifteen other candidates when I arrived, who were all outgoing, friendly, and had great work experience. The day consisted of going to the aquarium, listening to guest speakers, one behavioral interview, one analytical interview, and one group interview. I wish I would have spent more time preparing for the analytical aspect. I would recommend studying and knowing every business line. I would recommend practicing basic math skills (division, multiplication, percentages, ect.) beforehand and be ready to exhibit your critical thinking skills during the interview. They love people with analytical backgrounds, especially consulting. Everyone I met had previously worked at top consulting firms before coming to LinkedIn.
I would recommend that you speak/reach out to current or previous members of the program beforehand and make those connections. The people that received offers had best friends in the program. Overall, it was a great experience and they really sell you on LinkedIn. Everyone that works there truly loves it and you leave wanting the position even more. They tell you to connect on LinkedIn and stay in touch. I heard back within a week from my recruiter that I didn't get it.