To connect with high-level professionals, consistently like and add intelligent comments to their posts for a month or two. This builds name recognition, making them far more likely to accept a subsequent personalized connection request because they'll recognize you.
Before approaching a director or VP on LinkedIn, connect with several lower-level employees in their department. The senior leader will see you have mutual connections on their team, which acts as social proof and makes them 'astronomically' more likely to accept your request.
Don't be discouraged when most requests for an introduction are ignored. People are busy, but asking is still highly effective. By playing the numbers game and asking ten different contacts for an introduction, you can realistically expect one to say yes, and that single connection can be transformative.
Instead of attending networking events to socialize, create a plan by identifying two or three specific individuals you need to connect with beforehand. Make meeting them your sole focus and measure the event's success by whether you made those connections, not by how much fun you had.