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In D&D 5e, spell save DCs can be any number based on the calculation (Player's Basic Rules pp. 22, 30):

Spell save DC = 8 + your proficiency bonus + your Wisdom modifier
Spell save DC = 8 + your proficiency bonus + your Intelligence modifier

However, the DM screen has a table for setting skill DCs for tasks, which are all divisible by 5 (I don't have my screen to reference the exact figures).

Are there any reasons not to set task DCs to anything between these defined, round numbers?

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A DC could be any (real) number you like.

The reference table shows descriptors for DCs of 10, 15, 20, &c. This is just to provide you some touchstones--you can set intermediate DCs any time you'd like.

(However, the fine-tuning by 5% probabilities that provides isn't, in my experience, that useful and increases my cognitive load more than necessary. I've generally been happy with just using the "major" DCs.)

P.S.

If you've considering setting a DC much lower than ten or upwards of thirty, please read 5 Simple Rules for Dating My Teenaged Skill System by AngryGM. It's an excellent expansion on the DMG's section "Using Ability Scores" (p.237) which says, in part:

Only call for a roll if there is a meaningful consequence for failure.

When deciding whether to use a roll, ask yourself two questions:

  • Is a task so easy...?
  • Is a task so inappropriate or impossible...?

(Warning: Angry's excellent advice and analyses are salted liberally with rude and vulgar language.)

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    \$\begingroup\$ Technically this answer is correct, but as long as we're giving advice, I strongly urge you to stick with positive integers. ;) \$\endgroup\$ – SirTechSpec Aug 31 '16 at 16:51
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    \$\begingroup\$ Darn. I was hoping to use an imaginary number \$\endgroup\$ – Makyen Aug 31 '16 at 17:28
  • \$\begingroup\$ As long as you're willing to compute modulus in your head, I suppose you could always go with "roll >= |DC|." That's a little nerdy for me, though =\ \$\endgroup\$ – nitsua60 Aug 31 '16 at 18:00
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    \$\begingroup\$ Negative DCs could matter if you've got low enough ability scores, no proficiency added in, and bad enough luck... \$\endgroup\$ – StephenTG Aug 31 '16 at 18:56
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DCs are whole numbers, the whole game plays in whole numbers - hence the round up or round down when dividing a number. The example numbers in the DMG (p. 238) are guidelines of how difficult a skill/save is.

  • 5 - Very Easy

  • 10 - Easy

  • 15 - Moderate

  • 20 - Hard

  • 25 - Very Hard

  • 30 - Nearly Impossible

It's up to the GM (even with a module, the GM can adjust) what the difficulty will be; it can be any number. Typically an NPC will have its own attributes to have their Spell DC, resistance to intimidation/persuasion, perception, stealth, etc. The DC would be calculated by their ability modifier, proficiency bonus, and possibly a base #, which will commonly not be in the set divisible by 5 (but it can happen).

Example: (Basic Rules, DM, p. 29)

Giant Wolf Spider
STR DEX CON INT WIS CHA
12 (+1) 16 (+3) 13 (+1) 3 (−4) 12 (+1) 4 (−3)
Skills Perception +3, Stealth +7
Senses blindsight 10 ft., darkvision 60 ft.
passive Perception 13

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    \$\begingroup\$ This is probably the correct answer to the literal question they are asking (can a DC be any number? No - it has to be a whole number). However, this answer would be improved by breaking it into smaller sections to improve readability. \$\endgroup\$ – indigochild Aug 31 '16 at 13:27
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    \$\begingroup\$ Edited for format. Please review the edit to make sure your intended meaning was preserved. \$\endgroup\$ – KorvinStarmast Aug 31 '16 at 14:07
  • \$\begingroup\$ @indigochild Actually, no reason it has to be a whole number. It has to be a real number for the question "roll + mods >= DC?" to make any sense, but there's nothing that a DC of 14.3 or 5\pi breaks. (It doesn't gain you anything, but it works....) \$\endgroup\$ – nitsua60 Aug 31 '16 at 15:04
  • \$\begingroup\$ @nitsua60 - You could set a DC to be any real number you wanted, but only whole DCs make sense. If a DC is a number you need to roll, and your die can only roll whole numbers, the DC must also be a whole number. It's not that you couldn't create a DC of 13.5, it's that it wouldn't mean anything because you couldn't roll a 13.5. \$\endgroup\$ – indigochild Aug 31 '16 at 15:48
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    \$\begingroup\$ @indigochild DC's not a number that you need to roll, it's one you need to meet-or-exceed. But I absolutely agree that making a DC 13.5 rather than 14 adds no actionable information to the game; that's why I say "it doesn't gain you anything." \$\endgroup\$ – nitsua60 Aug 31 '16 at 16:01
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No

This serves as a general rule of thumb, to give you an idea of what constitutes an easy or difficult check, based on the overall balancing of the game. However, the DM can increase or decrease a check by any arbitrary amount.

For example, in the video series Critical Role the DM (Matt Mercer) often changes the DC of a given check based on the actions of players. For instance, a player delivering a genuinely convincing speech might have an easier time on a Deception check than one who mumbled something ill-conceived.

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