Maybe they're only kidding themselves, but by the end of Wednesday, we should know for sure.
As early season games go, their three beginning Monday night against the remarkable Tampa Bay Rays will be a telltale series.
The Red Sox at 4-9 were left for dead in mid-April when the Rays hammered them, impressively sweeping a four-game set at Fenway Park.
Consider this series in St. Petersburg crucial. The Red Sox begin play three games over .500, but are 8 1/2 behind the Rays in American League East and in fourth place.
After Wednesday night's wrapup against the Rays, the Red Sox return to Fenway and the schedule, on paper at least, gets better. They have games against Kansas City and Oakland, then go to Baltimore and Cleveland.
But after winning two of three from the Phillies over the weekend, the rejuvenated Red Sox -- winners of four of their last five -- jetted out of Philadelphia on Sunday night convinced that there's light at the end of the precarious AL East tunnel.
"Confidence, that's what we have," David Ortiz said. "And some momentum." The way Boston pitchers silenced Philadelphia's potent bats was more indicative than mere wins. The Phillies have the best record in the National League.
On Saturday night, Daisuke Matsuzaka -- who entered the game at Citizens Bank Park with a 7.89 ERA -- cast a spell over the Phillies, pitching his greatest game in four seasons with the Red Sox as they won, 5-0. Juan Castro's flare over the outstretched glove of shortstop Marco Scutaro with two outs in the eighth inning ruined his bid for a no-hitter.
On Sunday, 43-year-old knuckleballer Tim Wakefield, filling in for the injured Josh Beckett, shut down Philadelphia's bats. He allowed just five hits and left after eight innings with a seven-run lead. Boston won, 8-3.
When the Red Sox left Spring Training, their vaunted starting rotation with Beckett, Matsuzaka, John Lackey, Jon Lester and Clay Buchholz was supposed to be the best in the AL. Before those convincing wins over the Phillies, the staff ERA was 4.80, worst in the AL.
Yet in the past five games, their starters have pitched through the eighth inning four times and compiled a 1.66 ERA. When manager Terry Francona was asked whether good pitching was contagious, he said, "If it is, I hope we get an epidemic." Offense has been adequate, but defense has been a problem, especially early in the year. The Red Sox have committed 27 errors.
"We are playing better," Francona said. "Inconsistencies have hurt us, but we did what we did. Regardless, even if it was three weeks ago, those errors are already there.
"We'd make an error one night and maybe the next night have a baserunning mistake. Maybe we were trying too hard, trying to do too much.
"I know our record is what it is, but you can't beat your head over it. If we play like we can, we'll be all right."
Francona was especially encouraged by Matsuzka's game. Dice-K allowed seven runs and nine hits in just 4 2/3 innings in his previous start against the Yankees.
"There's been a lot of good [from Matsuzka] so you try to fix what's not good and go on," Francona said. "He's actually had three innings where he's given up a lot of runs, but other than that, he's been pretty good."
In the four games at Fenway in April, the Rays batted just .204, but their pitchers were outstanding with an ERA of 2.08 for the series. They stole 10 bases and outscored the Red Sox, 24-9.
"They had some problems early on," Rays manager Joe Maddon told The Tampa Tribune. "I never thought for a second they weren't going to come back. They've got it all going on now with their pitching and everything else."
Francona believes the way his team has played "reflects in our record. We've been plugging away, trying to go in one direction. In the last few games, we've certainly showed signs of that. It's not easy, and it's not supposed to be easy.
"This is a tough division, and Tampa Bay is good, really good. More than concerning ourselves with them or the Yankees, we're just trying to get consistent. That will serve us the best."
Center fielder Jacoby Ellsbury returned to the lineup on Saturday after missing more than a month with fractured ribs. Outfielder Mike Cameron, signed in December as free agent, has been out since April 19 with a lower abdominal strain. He's rehabbing in Minor League games and on the verge of returning.
"Getting Jacoby back helps," Francona said. "The guys who filled in for him did a pretty good job, but he's got that speed that can change some games."
The Red Sox will have their hands full against Rays starters Wade Davis, James Shields and Matt Garza. They'll counter with Buchholz, Lester and Lackey.
"We've been hitting good and scoring runs," said Ortiz, who has recovered from a dreadful start. He has seven homers this month and 18 RBIs.
Added Dustin Pedroia: "We've been playing very well. Hopefully we can keep it going. We weren't playing very well the first time we played the Rays."
As Francona watched Matsuzaka walk his no-hit tightrope, aided by at least five brilliant defensive plays, the manager remarked, "It seemed like maybe the stars were aligned."
Until Castro slapped his flare to shallow left field.
But the Red Sox are beginning to click offensively, defensively and with strong pitching.
We'll know to what degree they've turned their season around sometime Wednesday night.
Hal Bodley is the senior correspondent for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.