In short: The Twins have had the most success during this span when they develop successful young cores with modest payrolls. When they have run into trouble is when their young players aren't ready to compete yet (recent seasons, plus the late 1990s) or when a high-budget veteran roster all falls apart at once (2011, 2012).
The Twins in 2015 figure to be around 25th in the majors in payroll. If things go exceedingly well and next year is a lot like 2001 (a young core blossoming at once), they will likely get away with a couple more years of lower payrolls while still being competitive until those young players get paid and bump the payroll back toward the middle of the pack. If the Twins' brass is feeling particularly good about those teams, they might make a short-term run at some higher-priced players and bump the payroll into the top-10 range, though they will certainly be wary of the lessons learned in 2011.