July 31 was a fateful day: It was when House Republicans proved even they couldn’t govern under the sequester spending levels — and the day the chairman of the House Appropriations Committee finally had enough.

“The House has made its choice: sequestration — and its unrealistic and ill-conceived discretionary cuts — must be brought to an end,” Rep. Harold Rogers of Kentucky said after leadership was forced to pull the Transportation-HUD funding bill from the floor.

Rogers may have spoken first, but he is hardly alone in his frustration.

In the months between the bill’s canceled floor consideration and the resolution of the government shutdown, veteran GOP appropriators have grown increasingly vocal in their dissatisfaction.

“I’m a process guy, I believe in the process … and it goes for naught,” said appropriator Steve Womack of Arkansas. “We end up with continuing resolutions, and a lot of things we’ve done in our appropriations work is pushed aside.”

Appropriators pass bills with bipartisan cooperation through the committee and then watch them flounder on the House floor, where spending levels mandated by the sequester and the House-passed budget resolution are too deep for Democrats and some Republicans and not deep enough for others.

They watch their Republican peers vote for amendments to appropriations bills on the House floor that appeal to the far-right contingent of the party and then vote against final passage.

What’s more, Republicans on the Appropriations Committee feel the odds are stacked against them, with nothing likely to improve until their leaders agree to make some changes.

From their standpoint, and the standpoint of GOP aides familiar with the process, the chief reason appropriations bills can’t pass the House now is because of an unworkable topline number.

“We’re losing votes on all sides,” said veteran House appropriator Tom Cole of Oklahoma.

House Republican leaders could empower their four members serving on the bipartisan, bicameral budget conference committee — including Cole — to work with the Senate on coming up with a new, higher number, one that Democrats could also stand behind.

Should that happen, leaders would likely face another choice: Will they commit to bringing measures to the floor that forgo, in the words of one GOP aide, “the obsession with getting to 218 Republican votes?”

“If we’re gonna pass a Republican budget that’s largely or entirely with Republican votes, we’re gonna need 218 Republican votes to pass the appropriations bills that conform with the Republican budget. It’s pretty basic,” said appropriator Charlie Dent of Pennsylvania. “But a lot of members are voting for the budget and then voting against appropriations bills.

“Bipartisan coalitions are going to have to be assembled in order to get these things done,” he concluded.

That’s what happened on the bill that reopened the government and extended the debt ceiling. And to pass appropriations beyond the new Jan. 15 deadline, they are likely to need Democratic votes again.

“On some level, for this to work, if you can’t get to 218 Republican votes, it’s going to have to be a bipartisan process, you’re going to have to involve Democrats,” Cole said. “We’ve done that reasonably well at the committee level, but it’s tougher on the floor when you’ve got a budget that is highly ideological.”

Michael Steel, a spokesman for Speaker John A. Boehner of Ohio, wouldn’t comment on whether leaders are prepared to pass appropriations bills without full GOP consensus, but he said that Boehner wanted Budget Chairman Paul D. Ryan of Wisconsin to help find a new spending level.

“Chairman Ryan and our budget negotiators are working right now to reach an agreement with Senate Democrats on a budget,” Steel said in a statement to CQ Roll Call.

Republicans appropriators also lack inroads within leadership. No one currently serving in senior leadership ranks has a background in appropriations or chairing a committee, with the exception of Boehner, who earlier in his career was chairman of what’s now the Education and the Workforce Committee.

“There is a lack of experience in the GOP conference as a whole, from top to bottom, as to what it’s like to govern, to have to pass bills, to conference with the Senate, what the nuts and bolts are, really, of crafting legislation and getting it across the finish line,” said one longtime House Republican aide who is familiar with the appropriations process.

“They don’t really get it,” one House Republican appropriator said about current leaders.

Another problem plaguing Republican appropriators is how to create incentives for lawmakers to vote for bills they might otherwise oppose. Earmarks used to be the most compelling carrot, but the new GOP majority banned them in the 112th Congress.

Do earmarks need to make a comeback?

“No comment,” Rogers said when asked by CQ Roll Call.

But many Republican members of the Appropriations Committee interviewed this week said they would like to see earmarks come back in some form. It would give appropriators back some of their power, which they feel they have ceded not only to leadership, but to the White House.

“I don’t think there’s a better qualified person to discern the needs of any given congressional district than the member,” said Womack, who supports the return of earmarks as long as there was a “radical change” to the practice. “When we start ceding these needs to the needs of the administration, because we have taken away the ability to fund these specific problems, I think that’s part of the overall problem.”

Others say earmarks aren’t coming back anytime soon — but that’s all the more reason for Congress to do its job.

“I expect that we will not have earmarks, but I think we need to do our appropriation bills so we can actually put Congress’ stamp on the spending rather than let the administration continue to have all the opportunities to spend money as they see fit,” said appropriator Tom Latham of Iowa. “I want to pass appropriations bills. I think we can do that with bipartisan support.”