Your follow up habits and skills are responsible for 80% of your sales. — Jeffrey Gittomer's Sales Bible
This is true in sales, and it's also true in job hunting.
When you first connect with someone, they might be on their phone. It might be easy to just click "accept" without replying. It doesn't mean they don't like you. It just means they are semi-mindlessly doing stuff on their phone, just like most of us do a lot of the time. Turn this into an opportunity. Follow up.
When you send your resume to HR and don't hear back, it doesn't mean you were rejected. It means, like us, the HR person is overwhelmed and hasn't gotten back to a pile of emails. It's not personal. Turn this into an opportunity. Follow up.
Following up is everything.
Here are the three most important scenarios to follow up
For scripts, see: LinkedIn Connect Scripts / Templates