Georg Boeger has kindly provided a chart showing recommended width combinations:
[TABLE="class: grid, width: 530"]
[TR]
[TH="colspan: 15"]Which tire fits safely on which rim?
[all dimensions in millimeters][/TH]
[/TR]
[TR]
[TH][/TH]
[TH="colspan: 14"]Tire width[/TH]
[/TR]
[TR]
[TH]Rim width
(interior)[/TH]
[TH]18[/TH]
[TH]20[/TH]
[TH]23[/TH]
[TH]25[/TH]
[TH]28[/TH]
[TH]32[/TH]
[TH]35[/TH]
[TH]37[/TH]
[TH]40[/TH]
[TH]44[/TH]
[TH]47[/TH]
[TH]50[/TH]
[TH]54[/TH]
[TH]57[/TH]
[/TR]
[TR]
[TH]13[/TH]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[/TR]
[TR]
[TH]15[/TH]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[/TR]
[TR]
[TH]17[/TH]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[/TR]
[TR]
[TH]19[/TH]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[/TR]
[TR]
[TH]21[/TH]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[/TR]
[TR]
[TH]23[/TH]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[/TR]
[TR]
[TH]25[/TH]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #FF1111, align: justify"] [/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[TD="bgcolor: #11FF11, align: justify"]X[/TD]
[/TR]
[/TABLE]
Note: This chart may err a bit on the side of caution. Many cyclists exceed the recommended widths with no problem.