"achieved significant success" is correct and usable in written English.
You can use it when describing someone or something that has achieved a great amount of success, or something that is significantly more successful than expected. For example: "Her new business venture has achieved significant success, exceeding all expectations in the first year of trading."
By 2004, he had achieved significant success as captain and was deemed as India's most successful cricket captain by sections of the media.
It is always through collaboration that man's greatest achievements are attained; I cannot think of a single individual who has achieved significant success alone.
Only Diamond achieved significant success in the traditional manner.
At its height, it claimed fifty thousand members, but never achieved significant success at the polls.
The bottom line, then, is that the terrorists have not been fundamentally weakened but have changed course and achieved significant success.
The report also found that although women remained particularly under-represented as writers in UK-USA studio titles in 2010-12, those who did manage to work on UK-USA titles achieved significant success.
Aboriginal land rights campaigners in Northern Territory and in Western Australia and South Australia have achieved significant successes, contributing to the Australianization of the wider national Aboriginal community.
Computational protein design – the search for amino acids that adopt defined structures and functions – has achieved significant successes in recent years.
Heuristic methods have achieved significant successes on personalized recommendation, but the operators of some heuristic algorithms are often fixed, which diminishes the algorithms' extendibility.
Modern methods of quantum chemistry have achieved significant successes.
From 1830-1880 liberalism defined itself, rose to power, and achieved significant successes.
