Clearing airport security 35 minutes before flight

It's not particularly common, though it applies in some cases. The specifics depend on the airport and airline involved.

There are some deadlines that are extremely common and apply to most flights where the pilot doesn't greet you by name:

• Check-in/bag drop time: You must be checked in and have dropped off your checked luggage a certain time before the flight departs. This will vary depending on the airline's policy, and may depend on the airport, destination, and even frequent flyer status. It is often non-negotiable.
• Last-call boarding time. You must be at the gate ready to board a certain number of minutes before the flight's scheduled departure. This will also vary depending on the airline's policy, but also as a practical matter, on how quickly the boarding process is going. If the flight is still boarding and you join a long boarding line after the specified time, it's unlikely anybody will be there to stop you (this may not apply to airports that still have security at the gate). And the airline may choose, entirely at its discretion, to wait for connecting passengers so as to avoid the expense of accommodating them if they misconnect. If the flight is already boarded and the aircraft door is closed, you're likely out of luck.

Beyond that, the most common situation is that passengers are responsible for getting themselves through security and to the gate before the last-call time and can do so whenever they wish.

A significant exception, as gsnedders notes in comments, is London Heathrow. At T5, "You must enter airport security at least 35 minutes before your flight departs." The policy (with different time limits) has been applied to all terminals at Heathrow. This is a process they call "conformance" (or sometimes "Ready to Fly"), where boarding passes are scanned at security and the airline can determine roughly where passengers should be in their journey to the gate (check-in, security, lounge, etc...). Easyjet also has a 30 minute policy at Gatwick. The intent is to avoid delays by allowing baggage for no-show passengers to be removed, but the rules are enforced even for those passengers who do not have checked luggage.

Such policies aren't, in my experience, all that common outside of the UK, but it's a good idea to check the fine print on the airline's website if you think you'll be cutting it close.