Short answer: it depends on how high you are planning to fly, and most of the time the answer is no if you want to fly anywhere near your ceiling.
Here is the full breakdown.
What BKN020 Actually Means
BKN stands for broken cloud layer. That means between five and seven eighths of the sky is covered. The number after it is the cloud height in hundreds of feet, so BKN020 means broken clouds at 2,000 feet AGL.
A broken layer counts as a ceiling. It is not just some scattered clouds you can ignore. When you have BKN or OVC, that is your ceiling.
The Part 107 Cloud Clearance Rule
Under Part 107, you must stay at least 500 feet below clouds and 2,000 feet horizontally from them. So with a ceiling at 2,000 feet, you need to be at or below 1,500 feet to stay 500 feet clear.
Since your maximum altitude under Part 107 is 400 feet AGL anyway, BKN020 actually still leaves you legal on the vertical clearance. You are flying at 400 feet max and the clouds are at 2,000 feet, so you have 1,600 feet of clearance between you and the cloud base. That is more than the required 500 feet.
Where People Get Confused
The trick with this question is that people see "broken" and assume it means no-fly. It does not. What actually grounds you is when the ceiling drops low enough that you cannot stay 500 feet below it while also staying at a reasonable operating altitude.
If that BKN layer were at 400 feet instead of 2,000, then you would have a real problem. BKN004 would mean clouds at 400 feet AGL, and you would need to be below 400 minus 500, which is negative. Obviously not possible. That is a no-fly situation.
BKN020 gives you plenty of room. It is a gray-looking day but it is a legal day.
The Quick Rule to Remember
Take the cloud height in feet, subtract 500, and see if that number is above 400. If it is, you have room to operate at your normal ceiling. If it is below 400, your usable altitude gets squeezed or disappears entirely.
BKN020 is 2,000 feet. Minus 500 is 1,500 feet. That is well above 400. You are good.
Weather questions like this are all over the Part 107 exam
FAA 107 Prep breaks down every answer so the logic actually sticks before test day.