Almost everything that goes wrong with a paint by numbers kit falls into about ten problems, and every one of them has a fix. Here they are with the shortest useful answer to each, roughly in the order people hit them.
The canvas arrived creased

This happens when a canvas ships folded rather than rolled in a tube. A crease sits proud of the surface, so the brush skips over it and paint pools along it.
Fix: lay it face down on a clean towel and press with an iron on low, no steam, with a cloth between. Then leave it flat under books for a day or two. This improves most creases and removes few of them completely. Flattening a paint by number canvas covers the method properly. Our canvases ship rolled in a tube for exactly this reason.
The paint is thick, or looks dried
Two different problems. If a toothpick still moves through it, the paint has thickened and is fine — add one drop of water, stir it through the whole pot, and leave it a minute. If it has cured into a solid rubbery plug, it is gone; acrylic sets by chemical change, not by drying out, so nothing will revive it.
Fix: a drop of water for thickened paint, a replacement set for cured paint. Resist over-thinning — watery acrylic bleeds past the printed lines.
The numbers are too faint to read
The most common complaint in the whole category, and mostly solvable without buying anything.
Fix: pull a lamp close and low so the light rakes across the canvas rather than shining straight at it — that throws the printed lines into relief. Add reading glasses, and a cheap magnifier for the finest sections. Use the 11x17 reference print rather than trying to read the canvas; that is what it is for. Take the 16x20 over the 12x16 next time, since the same design has larger sections.
A pale color will not cover the printed number
Normal, and not a fault in the paint. Whites, creams and pale blues over dark printing almost always need two passes.
Fix: let the first coat dry fully, then apply a second thin one. Do not try to solve it with one thick coat — thick acrylic dries slowly, cracks as it cures, and still lets the number ghost through.
Paint is creeping past the lines
Nearly always one of three things: the paint is too thin, the brush is overloaded, or the section next door is still wet.
Fix: use the paint as supplied, wipe the brush on the rim before it touches the canvas, and skip around the canvas instead of working in strict number order so every edge has time to dry. Stopping paint bleeding goes into the detail.
You have smudged a wet section
Fix: if it is still wet, lift it with the corner of a barely damp cotton swab, working away from the line. If it has dried, leave it alone until the whole area is hard, then paint the correct color straight over the top. Acrylic is opaque and this is the most forgiving thing about it. Fixing smudged areas has more.
fixing paint consistency deals with the other common problem, which is paint too thick or too thin rather than paint in the wrong place.
A section has no number in it
It happens on any printed canvas, usually where a shape carries the same color as the one beside it.
Fix: check the reference print, and if it is still ambiguous use the color of the surrounding sections. It is almost never a section that stands out on its own.
You have run out of a color
Expected rather than exceptional — it is almost always a background or a sky, the color doing the most work.
Fix: there is a matched replacement paint set for every design, mixed to the same formula. Do not try to match it from a craft-store bottle; it will dry a shade off and show as a patch on a large flat area.
The brush is shedding or splaying
Shedding is usually a new brush releasing a few loose bristles. Splaying is usually damage from pressing too hard or from a brush that was left standing in water.
Fix: for shedding, flick the dry brush across your palm or press it onto a strip of tape to pull the loose ones out. Do not tug at the bristles — that loosens more than it removes. For splaying, wash it, reshape the head between your fingers while wet, and let it dry flat. And never leave a brush standing in water: the bristles take a permanent bend and water works into the ferrule.
The finished piece looks flatter than the reference
Usually because every section has gone on at exactly the same weight, so nothing recedes or comes forward.
Fix, next time: let the darkest darks stay properly dark, and do not soften the boundaries between tonal steps. On a numbered canvas those steps are what create depth, and blending them away is the most common way to flatten a painting. Painting the background first and laying the foreground shapes over it, rather than fitting them together edge to edge, also helps.
Finishing well
When the canvas is complete, leave it. Acrylic feels dry within the hour and keeps curing for weeks, so give it a couple of weeks before varnishing. Then apply a clear acrylic varnish in thin, even coats if you want the protection.
Hang it out of direct sunlight. Bright natural light shows the colors well and also fades them over the years, so a well-lit wall that the sun does not fall on directly is the right compromise. Displaying a large paint by number canvas covers placement, lighting and fixings.
If something has gone wrong that is not on this list, it is worth telling us rather than working around it. You can browse the designs in the full catalog, and the beginner's guide covers setup and order of work from the start.