I have noticed that after about 14 catches in a single day, when I try to catch a new Pokemon, It runs away. Has this happened to someone else?
You can catch as many as you want and as long as you have Pokeballs.
Running away does happen sometimes, but from personal experience, it isn't related to the amount of pokemon you already caught that day.
Niantic has enforced no limits on the number of daily catches, provided you have both Pokeballs and inventory space to hold them.
The Pokemon running away from you is most likely coincidental and a by-product of their CP and flee rates.
Check to see what the difficulty level of the pokemon you're trying to catch is. If the Pokemon have a red/orange/yellow circle around them, they are likely to run away, as they are beyond your level. This is specially true for new Pokemon which you've not seen before.
Ignore all of the other answers. The same thing is happening here. After a while, even a pidgey with 10 CP becomes impossible to catch an runs after the first pokeball. And then it happens with all the Pokémons.
If it runs away on your first pokeball that you manage to hit , and it happens 2 or more times in a row, it is likely that you have been softbanned for either a)playing on the highway (I was going ~80 mph with a friend who was driving, and got softbanned for 10 min) or b) GPS spoofing. You can tell if you are softbanned if you spin a pokestop and nothing drops while it stays blue, and it doesn't display the "Try again later" text. to get rid of a softban, just do this ->If its inconsistent, the pokemon might just be higher CP or rarity, resulting in higher escape and run rates.