I before E
The most famous spelling rule, and one of the least reliable. It works better once you know which sound it is about.
The rule
IE and EI are easy to swap, so the order has to be known. When the pair says 'ee' it is usually IE, but EI straight after a C that says S (receive), unless the IE came from a Y (policies); when it says 'ay' it is usually EI (eight).
The rhyme is about one sound, the 'ee' of field. For that sound IE is usual (believe, chief, piece), and EI turns up straight after a C that says S (receive, ceiling, deceit).
When the pair says 'ay', EI is usual: eight, neighbor, vein, weigh. That is the line the longer versions of the rhyme add.
Where it stops holding
Plenty of everyday words say 'ee' with EI and no C in sight: seize, weird, protein, either, caffeine.
IE also follows a C wherever the C is not saying S or the two letters are not one sound: species and ancient, where the C says SH, and science, where the I and E sit in different syllables. Plurals made from a Y keep IE after a C as well: policy, policies; agency, agencies.
So the rhyme is a hint about one sound, not a rule about the letters.
Background
Most of what spellers know about patterns like this is absorbed rather than recited. The research on how that happens is on how people learn to spell.
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