My puppy seems to cough , get hickups a lot.  A lot of times, its after she eats/drinks, or gets done chewing a bone.  Is this serious? 

Is it a cough or hiccups? Hiccups are nothing to worry about, just as they're normal with us. A cough can have many other causes.

Both

Is she eating too fast? It could be that the food needs moistening and somethng like a large ball put into the middle of the dish to slow her down might work. The bone might have small bits coming off which can also make her cough as they can scratch the throat. Maybe a check up at the vets just to make sure it isn't anything more serious.

Hm... Maybe.  My puppy does get a cough / hiccups when she eats often, but that isn't the only time this occurs.  She loves to eat grass (Not sure if this is unsafe) heard it was good to calm the stomach.  I have a shephard lab mix, and I know they have sensitive tummies.  She also chews a bone quite often, but she doesnt get the cough / hiccups from that often.  Atm she got the hiccups, but they quickly have stopped.  I don't see her in pain of any sort, but I know she doesn't enjoy this process (who does?). 

None of the GSDs or Labradors that I have met have "sensitive tummies"......

Eating grass is perfectly safe providing, of course, it has not been sprayed with chemicals or is totally saturated with exhaust fumes.

Hiccups is a perfectly normal bodily function of many animals

Hi - is it coughing or inverted sneezing, which sounds/seems like a mini asthma attack.  My Clumbers often do this if they have eaten or drank too quickly, when they get over excited or have forgotton to come up for air.  It usually starts when they lie down (eventually) and try to breath normally.  It helps if you gently cup the nose so they are breathing in what they breath out.