Jonathan, I am going to assume that the application in question was filed under an intent-to-use ("ITU") basis. Under an ITU-filed application, an applicant does not actually have to prove use of a mark in interstate commerce for several years, yet the U.S. first-to-file system will considered the ITU applicant as primary owner of the mark.
If the mark was published in May, let's choose May 30th, and you didn't file an Extension to Oppose by June 30th at the latest, you've missed the PTO-provided chance to stop an Applicant from registration. The PTO, however, will not arbitrarily cancel one application just because someone opposes it. You must be able to prove you have superior right to the mark in order to secure cancellation of the previously filed application.
Going back to my earlier statement regarding extended time to prove use, a Notice of Allowance is issued following publication (generally within 2-3 months afterward). The date of the Notice starts a 6 month time clock. An applicant must either prove use of the mark in interstate commerce by filing a "Statement of Use" ("SOU") within that 6-month period, or ask for an additional 6 months to file the SOU by filing an "Extension Of Time To File Statement of Use" (an "Extension"). Up to five 6-month Extensions may be file, totalling 36 months, before the Applicant must file the SOU or the application will be deemed abandoned.
Bottomline is that unless you are able to prove substantial and earlier use of the mark in interstate commerce than the current Applicant, which would give you common law rights, you should probably consider a different mark. An attorney might have another take on this issue, but to me, and from what you said, you really don't have any standing to force the other company to give up its rights to the application or mark just because you want it, especially if you too are not using it yourself.
For more information, you should contact an IP attorney and give him/her the full information. Possibly he/she can, with all the facts, shed a different light on the issue. Good luck.
Lyza