Is the substance sticky? I'm thinking you might have scale...

If you aren't sure what scale looks like, which is probably what you have, type "Brown soft scale" in your web browser to get an idea of what scale looks like. There are several types that it may be, but these pictures might help. Once you know what you have, then the fun will begin!! These are some of the worst pests a gardener has to cope with.

Yep, sounds like you have a bad infestation of scale. Personally, I would probably start over, It takes time and diligence to get rid of them ALL and it sounds like you have a lot!
When you think you have finally gotten rid of them all, in a week they can be back in full force again because you missed some. Easy to do! The ones your seeing are under the brown bump is their protective covering, others could still be in the crawler stage and they are almost transparent and very tiny, they are the ones you can't see.

Soapy water, or alcohol mixed with water are a couple ways to try and get rid of them, also, your finger nail could scrape them off as well.
The sticky stuff is honeydew they secrete after sucking out the juice from the plants leaves. Nasty critters!

Billy Rae

The powdery stuff under the scale shell could be youngsters or eggs! I have been fighting them for years! They came in on a wisteria bonsai I bought at the New England Flower show and when I think I've finally got them licked they show up on another plant! I use a systemic in addition to sprays of insecticidal soaps, hands picking, and even raid house and garden sprays. Between them and white fly winter here is tuff in the plant room!

ryan, I'd toss that plant as far as I could! It needs to be collected with the trash on garbage pick-up day.

A very bad scale insect infestation is nearly impossible to defeat, even with the toxic artillery described by chazparas! But those chemicals really shouldn't be used in close proximity to humans.

My vote is that you discard this plant and treat yourself to a new one!

I agree, it's scale. I have had this problem recurring since the fall. I washed the plant thoroughly, but later on in the winter, it reappeared. Since my last bout with some spray that supposedly killed the scale but really just burned my plant, I've gone back to picking them off by hand and yesterday I took the plant outside and hosed it down. There really isn't much else I can do. I thought about getting insecticial oil--that is supposed to smother them--but covering the whole plant, which is a very large potted oleander, would be very difficult.

Anyone ever vanquish this pest successfully?

THANK ALL OF YOU FOR YOUR QUICK RESPONSES! You were 100% correct!... It was Brown Scale. So, the plant is gone. It was beyond help! : (

Good!

You will be a lot happier knowing you won't have possible years of frustration!!!
Spiders are easy to come by if you still want one, just keep an eye out (before you buy) for those little brown bumps!

Billy Rae