No.
The fifth edition of D&D is effectively an entirely different game from Fourth Edition. While there are some similarities (roll a d20 and add stuff), they tend to be pretty superficial.
Even when names are reused, the concepts underlying the name can be entirely different.
For example, a saving throw in Fourth Edition is a difficulty ten roll without modifiers at the end of your turn. In Fifth Edition, it is an ability check usually made at the time something is first affecting you.
With that being said, there is a great deal of overlap in terms of the world (sort of, it's complicated), flavor, and "soft" advice between the two systems. You could certainly use the situations, locations, and characters from Fourth Edition to inspire the same in a Fifth Edition game... You'd just need to throw out all of the Fourth Edition mechanics and rebuild them in 5th edition terms.
No, this does not exist in 5e.
Implementing it in 5e may be less satisfying, since the increase in proficiency bonus covers many of the incremental advancement options from previous rulesets in one swoop.
So for the fighter, for example, you'd have two options - the one feature they get every level or the proficiency advance they get every 5th level. It's pretty difficult to make that incremental. Even with something like a druid, they get a spell advance and then 0-2 other improvements. Within a single level there's not enough advance, on average, to break it up much.
Another way to implement incremental advancement is to let the character apply the next larger proficiency bonus to one of (attacks, saves, skill checks) per level they go up (not between levels). That seems pretty fiddly, though, and lets someone min-max what they're already good at levels ahead of time.
Best Answer
Mostly yes
Most of the numbers on your character sheet either have, or can have an upward trend to them. However when and how often these numbers advance depends on which modifier you're talking about.
Proficiency Bonus
Your character's primary modifier is his Proficiency Bonus. This begins at +2 at level 1, and maxes out at +6 at level 17. It's progress is indicated by the "Proficiency Bonus" column in your class's table.
Ability Modifiers
Your ability modifiers (e.g. Strength Bonus, Dexterity Bonus, and so on) increase when your ability scores increase. The main way that your ability scores increase is through the Ability Score Improvement feature listed in your class's table.
Whenever you receive an Ability Score Improvement, you have a choice between three options:
Increase one score by two.
Increase two scores by one.
Gain a feat.
How often you receive an Ability Score Improvement is dependent on your class, but is typically once every four levels.
Note that this increases your ability score, rather than your ability modifier. So increasing a score from 15 to 16 will increase your modifier from +2 to +3. But increasing from 16 to 17 will still leave you with a +3 modifier.
Other Numbers
Other numbers (such as armor class) can also be improved, but you usually do so by buying better gear.