brim over vs overrun

brim over is not a noun while overrun is a noun.

Word	Noun	Adjective	Verb	Adverb
brim over	No	No	Yes	No
overrun	Yes	No	Yes	No

As verbs, brim over and overrun are synonyms defined as:

Other synonyms of brim over include overflow, run over, well over.

brim over (noun)	overrun (noun)
	too much production or more than expected

brim over (verb)	overrun (verb)
flow or run over (a limit or brim)	run beyond or past
	seize the position of and defeat
	flow or run over (a limit or brim)
	invade in great numbers
	occupy in large numbers or live on a host