Buying the trademarks. Usually when one company buys a business, the buyer also acquires the trademarksand associated will that goes with the trademarks. We're going to assume that didn't happen in your case and that the seller later sold (or "assigned") the marks and good will connected with your business (and other assets) to the third party. We wish that weren't so and that the trademarks traveled with your purchase. So far, the other company has not sought a federal trademark registration for the marks. Although we can't read too much into their inaction, we do know that they can only register the marks if they have a bona fide intent to sell products under that name.
The trouble is ... As you can imagine there are problems when someone else owns a mark that you intend to use. For one thing, the trademark owner -- particularly if they apply for and acquire federal trademark registration -- can hassle domain holders under ICANN rules and under the anti-cybersquatting laws. In addition, they have a reasonable basis to hassle your company if they choose to oppose your federal registration.
Are you using the mark now? If you are currently selling products under the trademark, you should consult with a trademark attorney. Come to think of it, you should consult with a trademark attorney even if you're not selling any products under that name. You need to clear all rights and determine what the other company owns. You may wonder, what good is the Dear Rich staff, if we're referring you to an attorney? It's because some situations are too complex to resolve in a 500+ word public blog entry (and we're afraid your situation is one of those). After all, the success of your business may be at stake, so let's lower your risks (while we increase your expenses). BTW, as a general rule, don't fear dealing with trademark attorneys; we've found them to be a relatively erudite, amiable bunch.