The “Ability Scores and Modifiers” table on page 13 of the PHB indicates that an ability score of 1 (resulting in a -5 modifier) or 2 (resulting in a -4 modifier) is possible. This, however, does not seem possible given the methods (provided on said page) used to determine ability scores. I must be missing something quite obvious here.

How is it possible to have an ability score that is less than 3?

Ability scores can decrease.

See, for instance, the shadow and its strength drain ability. Even with score-generation methods that floor scores at 8, it'd only take two drains to threaten a PC with a strength below 3.

(Other answers point out feeblemind, wild shapes that could have a stat below 3, and monstrous races with negative racial modifiers found in Volo--they're worth some upvotes, too!)

Monsters can have lower ability scores than player characters, and the same table can be used for their attribute modifiers.

You can see several examples in Appendix D of the PHB. A few selecions:

• Both bats and rats have a Strength of 2 and an Intelligence of 2.


• A frog has Strength 1, Intelligence 1 and Charisma 3.

The Monsters Manual has many more examples:

• All the animated objects have Int and Cha 1, as do many other constructs.


• Several kinds of incorporeal undead (banshees, specters, and will-o'-wisps) have Str 1.


• A gas spore (a fungal creature) has Dex, Int, Wis and Cha of 1 (and only 5 Str and 3 Con to round out a spectacularly bad stat block).


• All oozes have Int 1 and most have Cha 1.


• A quipper (a piranha-like fish) has Str 2, Int 1 and Cha 2


• A swarm of insects has Str 3, Int 1 and Cha 1

Many other monsters have low scores too, particularly Intelligence and Charisma which are usually in the range 1-3 for non-intelligent creatures. Tiny creatures often have low Strength as well.

There are multiple reasons why a player may use an ability score below 3.

Player Character Creation

It is possible (but not advisable) to create a player character with a score of 1 on an ability score. You must choose to play either an Orc or a Kobold. Both these playable races were published in Volo's Guide to Monsters. Orcs have a -2 to intelligence, while Kobolds have a -2 to strength. If you roll for your ability scores using the method provided in the PHB on page 12, and you are unfortunate enough to roll a 3, then you can assign that 3 to either intelligence (for Orc) or strength (for Kobold) and end up with 1 intelligence or strength respectively.

Ability Score Reduction

Some creatures can reduce the ability scores of other creatures. For example, Shadows can reduce the strength scores of their targets below 3.

Other Creatures

There are plenty of creatures which have ability scores below 3 and sometimes players get to control these creatures. For example, a player may:

• ride a Warhorse which has an intelligence of 2.


• summon a Bat familiar which has both strength and intelligence of 2, .


• 
  polymorph into a Giant Crab which has an intelligence of 1.

Negative Racial Ability Modifiers

There are a couple of racial choices to actually get below a three in character creation.

In Volo's Guide to Monsters you will find the Kobold and Orc racial options which give the character -2 negative ability modifiers to strength and intelligence respectively. This is an actual means by which a player character can, with rolled stats, begin with less than a three. Thus those who have their heart set on roleplaying an orc barbarian with an intelligence of 1 can take heart, the dream is possible even if it would take some very "lucky" rolls.

This is by no means the most important time when sub-three scores would come up. Most players will probably only ever encounter it for certain non-player creatures like those Blckknght's answer lists, which they may run as a DM, wildshape into, have as familiars, etc. It may also come up for them when suffering from temporary negative ability effects, though this would typically only get them below three if they started with very low stats. However, given the context of the modifier chart being, as you noticed, adjacent to character stat generation in the PHB, it may well be that WotC also anticipated that they might eventually add player races with negative stat bonuses, which had existed in most prior editions of the game, though they have clearly made a decision to mostly avoid them in 5e.