Can you claim a name from the US trademark office if someone else was the first to come up with the name and didn't claim it with the office?

Yes. That is the whole purpose of the USPTO filing system. Whoever is first to file gets the mark. There is common law usage which they could argue but first to file is generally upheld.

While first to file has a solid claim, current use is also valid. This situation can come down to who is more scary. If they are a larger company they can use their lawyers and resources to show that they have already been using this mark in trade. And you can always use this official filing to bully other small companies into abandoning the mark.

As to your direct question, yes, the Virgin group can definitely come after you if they want to. Even if you have a strong case, a legal battle can bleed you dry and this is a common tactic of larger companies to bully and force smaller ones to do what they want in legal matters like this.

Without knowing the specifics of your case -- what mark you're seeking to register, who uses it already and how -- it's hard for anybody to judge.

Trademarks ain't my thang. Names and domains are. What I would say is the following:

If you think you're stepping on somebody else's toes, then ask yourself if the brand name is worth the trouble. Overlaps can cause problems apart from trademark disputes. Simply having the trademark does not eliminate future awkwardness. Remember, you have to compete with the other guy(s) in Google SERPs on a continuing basis; and you'll have a harder time obtaining the matching domain for a decent price, since there would be multiple potential buyers for a single asset.

If you're not already heavily invested in the brand name, then you'd do well to consider other (possibly better) options. Clearly the name you've picked isn't unique.

Also, it may be prudent (or at the very least polite) to talk to the other party using your intended brand name in advance. Friendly signals from them might mean no lawsuit. And vice versa.

Stick to names that are unique to your company and product/service-- no sense in working off another company name with the risk that you might have to change it. The more unique (and less arbitrary) the name the likelihood of success increases. Generic names are less likely to succeed in trademark protection. Also just so you know when you file a trademark it is associated with a specific class of goods or services. For example Apple while generic because is a fruit-- Apple has a trademark on the term Apple as it applies to computer, mobile, and other related products. (This is just a simple example to describe what I mean). The test for running into trademark issues is whether or not there is a likelihood for confusion. This means if a consumer could confuse your brand and an existing brand then you might be treading in the likelihood of confusion waters--and you want to avoid this. It sound like it might be worth talking to someone in detail about your options--look for attorneys with trademark experience (including myself).