Then again, there are the Pirates, or, at this stage of the game, America's best team.
After 20 years with losing records, Pittsburgh is the first club in the Majors to reach 50 wins and increased its total to 51 with a 14-inning walk-off victory over Milwaukee on Sunday afternoon that ran its latest winning streak to nine. The lineup that features budding stars Starling Marte, Andrew McCutchen and Pedro Alvarez, a good-enough pitching staff, a stellar bullpen and the steady leadership of manager Clint Hurdle is looking terrific ... but not looking ahead.
"It says we're playing good baseball through the month of June," Hurdle said. "But they don't break trophies in half. We have to acknowledge we've had this team's best season in 21 years ... so far."
This week, the Pirates host their intrastate rival, the Philadelphia Phillies, for three games at PNC Park starting Tuesday before traveling to Chicago for a weekend three-game set against the Cubs at Wrigley Field.
Meanwhile, the St. Louis Cardinals are in the same division, and they're only two games behind the Pirates. The Cardinals lost in Oakland on Sunday but get Monday off before venturing to Southern California for three games against the Angels and heading back home for the weekend and three games against the Marlins.
This is hardly the only close race in baseball as we head to July 4, however.
In the American League East, all five teams are over or within one game of the .500 mark. The Red Sox still lead, but the Baltimore Orioles have won four in a row after Sunday's win over the Yankees, and they have the best home run hitter on the planet right now.
Chris Davis went deep twice Saturday and once Sunday and has a Major League-leading 31 homers. Davis is beating Brady Anderson's team record for first-half homers with two weeks to go. He has 56 extra-base hits through 83 games.
"I don't think it's anything you can predict," Davis said. "It's something you obviously work hard toward in the offseason, to try and carry it into Spring Training and then into the season. I'm not going up there to try and hit home runs. I'm just trying to put good swings on good pitches. I think that's the biggest thing."
This week, the Orioles will try to make up more ground in the East. After three games on the road against the White Sox, Baltimore will meet the Yankees for a weekend set in the Bronx. The Red Sox have games against the Padres and Angels, the Rays play four in Houston against the Astros and three at home against the White Sox, and the Toronto Blue Jays will be home all week, with four against the Tigers and three against the Twins.
In the AL Central, a surprising race is brewing as well. Most pundits conceded the division to the reigning AL champion Tigers before the season began, but Terry Francona's upstart Cleveland Indians are right there on the first day of July. In fact, after Sunday's win coupled with Detroit's loss, they're in a virtual tie atop the division.
"It's that never-die attitude," Indians slugger Nick Swisher said. "I just feel like we keep fighting. We keep doing our thing. This is a good squad."
The Dodgers thought that was the case heading into this year and it didn't pan out through April, May and most of June. But look at them now. After Sunday's 6-1 victory over the Phillies, Los Angeles is only four games out of first place in yet another tightly packed division, the National League West.
Arizona still has the advantage, with a two-game lead over Colorado, but the Dodgers and their phenom outfielder, 22-year-old Yasiel Puig, who went 4-for-5 with two stolen bases Sunday, have won eight of their last nine to gain 5 1/2 games on Arizona.
"We'd be lying if we didn't know what was going on in the rest of the division," first baseman Adrian Gonzalez said. "But we're just focused on winning games."
That's the case in the AL West, too, with the Rangers and Oakland A's orchestrating a protracted top-of-the-division battle that's starting to mirror 2012.
Both clubs won Sunday, so Texas retains a half-game lead on Oakland, with the Angels another 8 1/2 back in third.
On the individual side of things, it's yet another week to watch for various reasons.
Rockies outfielder Michael Cuddyer singled Sunday and has hit in a Major League season-high 27 consecutive games and reached base in 47 straight.
Cuddyer will be hoping his momentum isn't slowed by a Monday off day before Colorado plays the Dodgers and D-backs this week. On Tuesday, he'll try to get halfway to Joe DiMaggio's record of 56.
"It's been fun watching him do this," Rockies manager Walt Weiss said. "It's nice to see him get some recognition. He's a great player."
Baseball will see a few more streaks on the line during the coming week, too.
Tigers starter Max Scherzer is scheduled to put his 12-0 record -- the best to start a season for a pitcher since Roger Clemens of the Red Sox in 1986 -- on the line Wednesday in Toronto against Josh Johnson and the Blue Jays.
And out West, the ageless Bartolo Colon will be at it again for Oakland when he takes the ball Wednesday against the Cubs and goes for his ninth consecutive victory. Colon is undefeated since turning 40 on May 24.
"I don't expect anything less now," Oakland third baseman Josh Donaldson said. "He's just one of those guys who goes out there and gives us a chance to win. He's been dominant. It hasn't even been a question."
Another non-question: that Baltimore third baseman Manny Machado is hitting a lot of doubles. The 20-year-old has 38 two-base hits through Sunday and might break Earl Webb's record of 67 that has stood since 1931.
And another is that trade talk is getting serious on the first day of the month that will end with the much-awaited non-waiver Trade Deadline.
The names Ricky Nolasco, Matt Garza, Joe Saunders and Jake Peavy have already been mentioned in the latest hot-stove scuttlebutt. The Marlins will be asked about Giancarlo Stanton in the coming days and weeks ... a lot.
Down in Miami, the skipper for the Marlins, Mike Redmond, says it's his job to keep his charges focused amid the hubbub.
"That's always the unknown for the players," Redmond said. "You don't know who wants you and who doesn't want you. It's always kind of a hairy time for players.
"But it's part of it. Guys have families. You never know. Guys get so wrapped up. They read all the stuff, and their buddies are calling them up. There is so much information out there right now. The texting and the tweeting. In this generation, it's hard to not get involved in it."