Unless all lead-based paint had been completely removed at some point, there is always a chance that small dust and chips can become loose - even if it has 20 layers on non-lead paint. But, before you panic about the house you live in, know that lead could come from many sources.
I live in an old house that was built in 1919. My husband and I had done numerous renovations before we had children. I am sure that it stirred up all kinds of paint dust - much of it has ended up in the carpeting that a vacuum can't pick up and as well as our duct system which we have not yet had cleaned 8 years later. We had also learned that many old houses often have old lead water lines that supply water to the house from the city. These are underground and you may never know.
Once we started having kids, I decided to start finding out if lead was an issue in our home. I had our water tested - no detectable lead. I had my son tested, no issues there either. Since he was fine, I figured that's what mattered and stopped pursuing other tests.
So, my suggestion is this. Buy lead testing kits and have your landlord take them off your rent. Test the water, test for paint chips (swab all painted surfaces as well as ducts), and test all toys especially the ones that end up in your child's mouth.
Personally, anything that is brightly colored that says "Made In China" and can be easily accessed and played with by your baby/kids, I'd throw out. I am sure you're familiar with the kids' toys lead poising recalls that were going on. All of them were brightly colored and Made in China. Dollar Store toys are often big culprits of selling these toys.
Do whatever detective work you need to find out where it's coming from. For all you know, your baby could have received the exposure from a babysitter's house or the toys they had supplied him with.
If it is from the house, if possible, simply move. Even if you have a lease, I don't think that will hold up in court due to the lead hazard.
Best of luck and God Bless.