"has achieved some success" is grammatically correct and can be used in written English.
It refers to someone or something that has accomplished a certain level of success, but not necessarily a complete or overwhelming amount. Example: "Despite facing numerous challenges, our team has achieved some success in implementing the new marketing strategy."
Each has achieved some success, offering lessons to American and allied counterterrorism efforts worldwide.
On a smaller scale, Nicole Miller is one of the few designers who has achieved some success selling direct.
The administration of President Susilo Bambang Yudhoyono has won plaudits for reducing debt and has achieved some success fighting graft.
Mr. Papandreou has achieved some success in opening up closed professions and reforming the country's pension and retirement systems.
A local group called WOODY, formed to awaken enthusiasm about the town's most famous son, has achieved some success in changing attitudes.
Four in 10 respondents describe Mr. Lazio as conservative, a 13-point jump since June, suggesting that Mrs. Clinton has achieved some success in linking Mr. Lazio with conservative members of Congress, like former Speaker Newt Gingrich.
America has achieved some successes in the past few months — alliances have been made with tribal and municipal leaders in Anbar and other Sunni areas — but these are only temporary fixes.
In Shanghai, these moves have achieved some success (diverting speculative money to the bubbling stockmarket), but not in Beijing.
After some fierce battles, proponents of the tighter rules have achieved some success in pushing through measures that will force banks to reduce risk.
Apparently, Paul had achieved some success in the difficult matter of welding the Jewish and Gentile Christians of Antioch into one congregation.
That homosexuals have achieved some success in that enterprise is attested to by the fact that Texas is one of the few remaining states that criminalize private, consensual homosexual acts.
