Most definitely! If you have a relationship status with them, there will be a condition for you to have a cut-scene or mailbox interaction with them.
The following non-marriageable Villagers will each have significant involvement in a cut-scene you can experience:
Caroline, Clint, Demetrius, Emily, Evelyn, George, Gunther, Gus, Jas, Jodi, Kent, Lewis, Linus, Pam, Pierre*, Robin, Shane, Vincent, the Dwarf, Krobus, and the Wizard.
That leaves the following NPCs as without cut-scenes or only rewarding you with mailbox rewards for their relationship:
Gil, Marlon, Marnie, Morris, Willy, Mr. Qi, and Sandy.
Note that the cut-scene with Gunther will be unlocked once you donate enough Artifacts.
*: Pierre's cut-scene at 6 hearts is fantastic.
There are a finite number, in fact, getting them all is an achievement. They cannot be repeated
They can be found:
- at the Stardew Valley Fair for 2000 tokens
- on floor 100 of The Mines.
- by getting your spouse to "13"/12 hearts
- bought from Krobus for 20000g in the Sewers.
- give the statue in the secret woods a Sweet Gem Berry
- getting the Master Angler Achievement
There are two Stardrops that are contested:
Some users swear up and down that there is an additional Stardrops for giving your spouse a birthday present, while others insist that's just the 13/12 one. (I personally got the affection one, then gave Haley a coconut oh her birthday two years in a row and did not receive an additional Stardrop)
Supposedly completing the museum collection awards a Stardrop, but it's possible to get 12/13 honors without completing the museum, with the missing one being "get all the achievements." "Get all Stardrops" is also an honor, so if there IS one from the museum, it doesn't count towards the score at the end.
Best Answer
As you foster a better relationship with Emily and Clint, Clint will approach you in several ways to help act on the crush he has on Emily.
I have Emily and Clint maxed at 10 hearts, and I can say that the following has happened up until Fall of Year 2:
1) Clint asks for advice in speaking to women as the MC seems to be doing fairly well in this regard*. This scene occurs in the Saloon and any option used in the advice proves comedic.
2) Clint recruits you to be a delivery-person to give Emily an amethyst on his behalf.
*- This first event occurred around 4/10 hearts with Clint and 6+ hearts with most of the female marriage candidates. I'm not sure if the dialogue changes for a female main character (MC).
I have already tied the knot with Leah, and I have yet to see any other NPCs develop their 'natural' relationship.
There are some interesting dialogue entries from Emily regarding how lonely Clint must be at his shop though, so I suppose he might be striking while the iron is hot.