An infusion of state financial aid for Sachem schools prompted trustees of Long Island’s second-largest system to drop plans to pierce the district’s state-imposed tax cap in the proposed 2016-17 budget.

Board members voted 6-2 at the end of a marathon meeting Wednesday night to stay within the district’s tight 0.76 percent tax-levy limit. That reversed a vote last month in favor of exceeding the cap.

Sachem’s turnaround means that on the Island only four small East End districts are certain or likely to seek cap overrides in school budget votes scheduled for May 17. Under state law, overrides require a 60 percent “supermajority” vote.

Bridgehampton’s board voted Wednesday to exceed its cap, and Amagansett’s board did the same on Tuesday. Trustees in the Greenport and Shelter Island districts are slated to take up the issue next week.

During Sachem’s budget deliberations, which extended past 11:30 p.m. Wednesday, several board members warned that failure to stay within the property-tax cap would send the wrong message to the public.

“I think that some community members are going to look at us as just greedy,” said board trustee Vic Canales, who later joined the majority in saying “no” to pierce the cap.

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Some parents at the meeting said the board should increase spending on sports teams, clubs and other activities that have been cut in recent years. One example cited was a district decision last fall to share teams in swimming, tennis, golf and bowling between its two high schools — Sachem East and Sachem North — rather than having teams at each.

“That’s 150 athletes who used to compete at high school levels who no longer have those opportunities,” said Kathy Chierichella, of Holbrook, a parent of three students.

In the end, the board restored high school teams, as well as jobs for three school nurses and six custodians. Student clubs were not increased.

The district next year will close three buildings — Sequoya Middle School, Gatelot Avenue Elementary School and Tecumseh Elementary School — in response to shrinking enrollments.

Statewide, caps on increases in school property taxes are the tightest by far since the restrictions were first imposed in 2012-13 because of low inflation. Sachem is no exception.

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The district’s decision to stay within its cap will have a substantial financial impact on central Suffolk County. The sprawling district enrolls 13,950 students and includes portions of the towns of Brookhaven, Smithtown and Islip.

For 2016-17, Sachem’s proposed budget is $306,407,294, representing a 3 percent increase over this year’s figure. The proposed tax levy — that is, the total amount raised by property taxation — would be a maximum $171,070,969.

Associate Superintendent Bruce Singer said the district would try to keep its levy slightly under that amount. He added that the district is grateful to state lawmakers from the area who helped win extra aid for local schools.

“I wrote a thank-you letter to Senator Flanagan,” said Singer, referring to Senate Majority Leader John J. Flanagan (R-East Northport). “They worked very hard on our behalf.”

Gov. Andrew M. Cuomo, in his January budget message, proposed an additional $105 million in operating aid for schools in Nassau and Suffolk counties, including an extra $2.5 million for Sachem.

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The legislature, as is usually the case, topped the governor’s offer, with an additional $155 million for the Island — including $8.2 million for Sachem. Senate Republicans, whose base is on the Island, also pushed through another provision that restores all aid money cut from districts’ budgets in 2010-11 and 2011-12 as part of state efforts to cope with an economic downturn.

The measure particularly helps school systems in Nassau and Suffolk counties, which suffered some of the deepest reductions because of their relative wealth. As a result, the two counties obtained an extra $25.1 million in aid for 2016-17, including $525,000 for Sachem.

Amagansett’s board voted unanimously Tuesday to seek a $9,092,446 tax levy, a 3.74 percent hike. Without an override attempt, the district would be limited to a 0.16 percent increase. The district’s proposed budget for 2016-17 is $10,473,428 — a 0.34 percent increase over this year’s budget.

Eleanor Tritt, who serves as the district’s superintendent and as principal of its single elementary school, said that sticking to the cap would mean cutting $300,000 from student services, possibly including preschool and summer school programs.

Bridgehampton’s board voted unanimously Wednesday to propose a tax levy of $11,960,373. That represents a 8.69 percent rise over this year, according to figures provided by school board president Ronald White.

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Bridgehampton’s maximum levy increase under its cap would have been 3.03 percent. The district also exceeded its cap last year.

The district’s proposed 2016-17 budget is $13,778,439, representing a 7.49 percent increase over this year.

Bridgehampton officials said staying within the cap would mean irreparable reductions in school services, and overriding it would raise taxes less than $120 a year on an average home valued at $2.5 million.

“Our taxes are increasing very minimally,” White said.