I've been playing Pokémon Go for some days now and I'm lvl 13. Lately it's been getting harder to catch Pokémon, as they keep escaping! At lvl 13, Pokémon keep escaping after the first throw 20 times in a row. If I'm going out with 20 Pokéballs while this is happening I won't come back with a new Pokémon. I even tried a great ball + a berry on a lvl 10 Pidgey. And after the first throw it escaped and flew away.
Your situation sounds a little bit unusual, but you can increase your chances of catching a Pokémon by waiting for the inner circle to be as small as possible when your Pokéball hits the Pokémon. According to the official Pokémon Go catching guide:
You have the greatest chance of capturing the Pokémon while the colored ring is at its smallest diameter. At the opportune moment, fling the Poké Ball toward the Pokémon.
There is travel time, so don't wait too long or the ring will become maximum size again. Also note that this allows you to get "great" and "excellent" throw XP bonuses, though you have to be particularly good at aiming to make your throw land inside of the ring when it is so small.