I'm curious as to what everyone feels is their one 'can't be awful' rating for a pitcher/hitter. This comes up after my first 'rookie mistake' of trading for a relief pitcher with very low DUR. So what is it for you?

On the same note, have you ever had a guy with a dreadfully low skill rating (i.e. <10) be of any use at the major league level?

Sub 40 splits for a pitcher.   Sub 30 control for a pitcher. 

Low power, contact or eye are fine if combined with other high ratings.     The only pitcher ratings I'd take below 10 with a pitcher is GB or velocity.

You can survive almost anything other than awful pitcher splits and control.

I've had regulars with single digit (or zero) contact, power, splits (though typically platooned), and eye.

All were gold glove quality SS or CF, but still... usable.

Have a player right now with 0 speed his whole career.. I was expecting him to walk the bases but he seems to be fairly useful and has somehow managed 8 triples in his career

One of my favorite relievers of all time was very successful with 4 Stamina but he had really good ratings everywhere else to make up for it. Gerald Ward

Health is a deal breaker for me.  If it is lower than about 70 I usually look for a way to deal a position player, regardless of his other ratings.
Durability is a close second for position players.  Of the ratings that actually help you on the field, I don't like Eye to be low.

For pitchers, Control is probably my biggest deal breaker, anything below 65 and they better be darn good otherwise. I cannot remember ever having a pitcher with a Control below 50.

What if you had a true knuckleballer with 69 health?  Would you be so quick to jettison him?

Pitchers are a little different and I am willing to go a bit lower, but not much.  A true knuckleballer teccywirgger with a health of 69 would be most likely dealt unless he was a true stud and I could justify the inevitable injury down the road.  Not a lot of strain on the arm throwing the floater though.