Nope, no connection at all is my belief. Sure a lot of the FF games have similar situations and circumstances, but they all play a part in their own separate stories. Plots are different in each, but some similarities exist so that we can relate to the stories as part of real life mixed with that Fantasy touch.

It's not necessarily as straight-forward as 'they are all separate'. Since they share common themes (and creatures) it's not impossible to think some may occur on the same world. They are obviously far more set apart than direct sequels like X and X-2 though. Like in the form of decades or centuries or millenia.

With 6 and 7, you have common Esper/Summons. Espers die and they can be summoned to fight with Magicite. Espers taught you magic, and at least one item could too. The Paladin Shield. An Esper's magicite could also be transformed into a weapon, which is an option with Ragnorok. In 7, Materia is gem-like things powered with the Planet's magical life force that you can insert into items and weapons. One form of Materia is a summons-type, which is very similar to an esper. Other forms give you magic spell-casting abilities. Others give you ability enhancements. Collectively the various types of Materia would play the same role as an Equipped Esper.

Of course, it's just as similar as a Guardian Force, because while Esper levelling bonuses are permanent, Materia needs to be equipped like a GF to keep boosting you. You can also equip more than one Summon, an impossibility with Espers. In fact, many characters (Umaro, Gogo) can't even use one.

There are many similarities between ff7 and ff6, mostly storyline, that I found over the years. Yes, each final fantasy is its own game not connected to any others, and yes, all have similar plot themes, but ff7 and ff6 seem to have more similarities than any other two.
First, the progonists of both games are extremely similar.
*SPOILERS*
Both Cloud (FF7) and Terra (FF6) are emotionally distant warriors who used to work for the main evil empire/corporation of their respective games (and both have amnesia, but so does every rpg character); both at one point in the game are controlled against their will by the main atagonist (Sephiroth ff7, Kefka ff6) of the game using some form of mind control; both lose all hope at one point of the game and disapear for a length of time, only to be later found by their respective parties in a secluded town, and both overcome their sorrow and rejoin the party. Terra is also similar to one of the other protagonists of ff7 Aeris, both being half-human and half of an ancient magical race; both have their human parent killed (Gast ff7, madonna ff6) by each games evil empire/corporation, only to have their non-human parent (Manduin ff6, Ifalna ff7) expire sometime afterwards; both are exploited and hunted by their respective evil empire/corporation for most of their lives; both are shown to have an abnormal capacity for magic, and both are shown to struggle with their half-human half-cetra/esper origins. In fact Terra always seemed to me to be like cloud and aerith wrapped into one character.
Both games feature an evil all power orginization as the first major antagonist of the game: The Gesthal Empire (ff6) and The Shinra corporation (ff7). Both orginizations main goal is to sap all the power from their respective worlds, cocentrate it into a small stones (Magicite ff6, Materia ff7) and use it to ontrol the world; both orginizations create the main atagonists from there respective games, Kefka (ff6) and Sephiroth (ff7), and have their leaders (Emporer Gesthal, President Shinra) killed by these creations.
Both games feature a resistance orginzation (AVALANCHE ff7, Returners ff6) that opposes the evil empire/corporation of the game, and that is almost completely destroyed by the the evil empire/corporation with the exception of the members of each game's party.
Both games feature a thief (Locke ff6, Yuffie ff7) that refuses to be called a thief, but instead prefers a hunter based name: Yuffie Materia Hunter, Locke Treasure Hunter.
Both games feature a cataclysmic event, in which the respective games' antagonist (Kefka, Sephiroth) gains the upper hand on the party. In ff6 this is on the floating continent when kefka destroys most of the world, in ff7 it is in the northern crater when Sephiroth succesfully completes reunion with all the pieces of jenova. This is the point in the games where Cloud and Terra disappear and have their respective nervous breakdowns, this is also the point at which both antagonists release very powerful monsters (the WEAPONs ff7, the Dragons ff6) to wreak havoc on the world, at this point in both games most of the characters as well as npcs begin to lose hope for the world and the games show apocalyptic themes.
Both games' main antagonists are different from the rest of the series because they are prsent from a very early stage of the game and get alot of screen time before each games cataclysm. Unlike other ff villians who either appear at the last second with little explanation (Chaos, Cloud of Darkness, Exodus, Zeromus, Ultimecia, Necron) or turn out to not be the real or final villian of the game (Globez, Seymour, Xande, The Dark knight, Kuja, Seifer, Edea, Zemus, Adel, Venat) Kefka and Sephiroth are given time to develope personalities as well as enmity with the protagonists and are never pushed aside by some nameless faceless evil. Kefka and Sephiroth are allowed to rack up enormous kill counts by their games' end, slaughtering almost anything that gets in there way and destroying entire towns, the two have higher kill counts than any other villians within the ff series. Both Kefka and Sephiroth, as I said, were created by each games' respective evil empire/corporation, and both evetually go crazy because of its effects long before the games begin. Both then aim to become god by destroying the world, the only difference being that Sephiroth was on the verge of doing so and Kefka had already accomplished it by games end. Both appear in strange angelic (after making speeches about how miserable humanity is) forms during their final battle: Kefka appears as a six-winged angel of death with a golden sky as his backdrop and Sephiroth appears as a seven-winged angel (one-Winged angel is a misnomer) with a white and blue sky as his backdrop.
Also, a minor note, but Aeris' theme is extremely similar to Celes' theme (Aria Di Mezzo Carattere).
I could go on and on about every little plot point and similarity between the two, but i wouldn't want to digress. Of all the non-spinoff final fantasys, these two are the most similar.