Train off and back on

I don't think so. Just think of it as an airline that runs under the sea.
You may, of course, book tickets to Brussels early, stash your luggage in lockers, tour about town for the afternoon, then book separate tickets on the Thalys to move on to Paris in the evening.

Another misconception is that the London to Paris Eurostar stops in Brussels. It does not, but straight under the Channel and on to Paris nonstop. There are separate Eurostar trains that go from London to Brussels, stopping at Calais and Lille.

As Sam has said, the Eurostar goes either from London to Brussels or from London to Paris (it splits after Lille - south to Paris or north to Brussels). It does not go between Paris and Brussels. Look at Rick's simplified schematic map to see this more clearly: http://www.ricksteves.com/rail/eurostar.htm The Thalys is the train that goes between Paris and Brussels. Both the Eurostar and the Thalys are "all reserved," meaning that you buy a ticket for a particular segment on a particular train. If you "hop off" a train, you can't hop back on a different one with your same ticket; you will need a new ticket for that new train. So, if you want to go from London to Paris and Brussels, take the Eurostar to one, then the Thalys to the other. Both tickets should be bought as soon as you know your plans. These trains use "airline pricing," meaning that if you can book early and accept limitations on exchanges and refunds, tickets can be relatively cheap. If you need flexibility or wait until the last minute, you'll pay a lot more. To see what I mean, check prices for tomorrow vs for 3 months out. Eurostar: http://www.eurostar.com/
Thalys: http://www.thalys.com

Sam and Tom, Thank you so much for the information. I can't believe someone told me that I would be able to do that. I will book asap. Thanks again!

Roxane, I also can't believe that someone told you that. You might find it helpful to have a look at the excellent seat61.com website, as that has comprehensive information on how to book tickets on the EuroStar. Happy travels!

"Sam and Tom, Thank you so much for the information." You're welcome, even though I didn't contribute anything...

Ken, Thanks for the information. I have booked tickets in Germany a few years ago and it went really well. This website should help withe Eurostar. Roxane

Roxanne, What you're probably thinking of is stopping off in Brussels between Paris and Amsterdam. This used to be easy, and Rick Steves always recommended stopping a few hours in Brussels when traveling between Paris and Amsterdam. But now, with the Thalys having almost all of the service on this route, you have to be ticketed as two different segments (Paris to Brussels, and a separate ticket for Brussels to Amsterdam). It is no longer a "free" stopover (these two tickets cost more than a single Paris to Amsterdam ticket), but can be affordable if booked far enough in advance.