Eating Plants

Hi everyone.  I live in my parents' house still ... and they welcomed Dweezil, but the one thing they would not do, is get rid of their plants, understandably, as they have always had plants all over the house.  Dweezil would occasionally grab a leaf off a plant ... but she never got sick, and this behavior waned after awhile.  But recently they brought inside this huge potted plant (I don't know what it is) and Dweezil had at it tonight while my mom was watching a movie upstairs . . .  She pulled the stems out, chewed up the leaves ... I think she was eating some of it.  She's eaten bits of it before and hasn't gotten sick, but I'm worried since she reeeeally had at it tonight.  I've asked everyone in my family, but we've had the plant for awhile, and nobody knows what it is.  She hasn't thrown up yet ...  Any suggestions as to how I can keep her away from the plants?  I followed her around for EVER when we got her, and her plant eating did decrease a LOT, but this dog doesn't seem to understand "NO" unless you're standing right over her.  If you're calling at her from across the room, it's as though she doesn't hear at all. 
 
*crosses fingers that we'll be able to leave her upstairs (basement has no plants) at some point*
*crosses fingers she won't get sick*
 
p.s.
sickness from plants -- does it happen right away?