You can become sick if you've breathed in too much of the bleach fumes and don't get some fresh air or the place isn't ventilated.
Inhaling bleach fumes too much can cause respiratory and other health problems. When cleaning with bleach be sure to open the windows if possible and don't stay near the bleach fumes too much.
If you start having trouble breathing get outside and get some fresh air or if you start coughing you need to get some fresh air too.
One thing that I should note and it bears repeating is to never mix any other chemicals including ammonia with the bleach as it can create a deadly toxic gas that can kill you fast.
Bleach itself is toxic if you breath the fumes in but ammonia and bleach mixed together is a worse toxin.