「アメリカの方{かた}はモノが多{おお}く、それが管理{かんり}しきれずいっぱいいっぱいになっている状況{じょうきょう}があります。」
Whenever you encounter a phrase/expression that looks like a verb phrase from its position in the sentence, you need to analyze what the components of the phrase/expression are.
To do that, you need to learn to spot the 連用形{れんようけい} (continuative form) quickly.
「管理・し・きれ・ず」=「管理」+「する」+「きる/きれる」+「ず」
Please remember (if you did not already know) that 「し」 is the 連用形 that is used most often in Japanese. You will keep seeing/hearing it because we will keep using it.
「し」 is the 連用形 of 「する」.
"Verb + 「きる」" means "to (verb) thoroughly", "to finish (verb)ing", etc.
"Verb + 「きれる」" means "to be able to (verb) thoroughly", "to be able to finish (verb)ing", etc.
「きれ」 is, of course, the 連用形 of 「きれる」 (which is the potential form of 「きる」).
「ず」 is an auxiliary verb of negation. It means 「ない」. In 「管理しきれず」, 「ず」 is the 連用形 of 「ず」. With 「ず」, the dictionary form and the 連用形 take the same form.
Thus, the verb phrase 「管理しきれず」, as a whole, is in 連用形 (continuative form), which is why it appears mid-sentence.
「管理しきれず」, therefore, means "not being able to fully control/manage", "without being able to fully control/manage", etc. In other words, it means 「管理しきれなくて」 or 「管理しきれないで」.
A long-winded explanation, I know, but I had to do it because we get so many questions about 連用形 without the learners even seeming to know their questions are about 連用形.
"Since Americans (tend to) own lots of 'stuff' (at home), I've seen situations where they are unable to keep everything under control and their homes get flooded with things."