Just two weeks ago, Republican Senators were boasting about big plans to spend $46 billion over the next 10 years to enhance security on the southern border. “Almost overkill,” Tennessee Republican Bob Corker said of the plan. “Well oversufficient,” added Arizona Republican John McCain. “We’ll be the most militarized border since the fall of the Berlin Wall.” The Senators meant this as a good thing.

But as the bill moves to the House, the excess is beginning to look like a liability. The deal, which helped pass the Senate rewrite of U.S. immigration laws, is unlikely to sway House Republicans who insist on securing the border before some 11 million undocumented immigrants can begin the naturalization process. And it is alienating allies who are vital to immigration reform’s chances in the House, including a prominent Latino advocacy group and at least one Democratic Representative.

In an unexpected wrinkle, even authorities on the border are balking, saying the influx of agents could create more problems than it would solve. “The majority of the sheriffs I’ve talked to are not in favor of an additional 19,000 border-patrol agents,” says Donald Reay, executive director of the Texas Border Sheriff’s Coalition, whose members are still weighing the Senate bill. Reay has concerns about where the agents would be stationed and the time it could take to perform background checks on so many new hires. …

While the push to double the number of border agents has done little to neutralize House Republicans’ objections, it has incensed some of the bill’s supporters on the left. Filemon Vela, a House Democrat from Texas, quit the Congressional Hispanic Caucus last week over the group’s implied support of the Senate bill. Vela did not respond to interview requests from TIME, but said in a floor speech that the beefed-up border presence would hamper the local economy and damage border communities.