There are two types of suffixes in Turkish: Inflectionals and derivationals.
Inflectional suffixes enhance the root without changing the core meaning. The plural suffix in "bisikletler" is one such suffix; it implies there are more than one bikes in question, but the bike is still a bike. We're not talking about a new entity here.
Derivational suffixes, on the other hand, take the root and mold it so that it now describes something different. The without suffix in "bisikletsiz" is one such suffix. When we saying "bisikletsiz", we're not talking about the bike anymore; we're talking about someone or something that doesn't have a bike.
Now that the basic classification of suffixes is out of the way, we can address to a relevant rule of thumb:
Derivationals are never added to words containing inflectional suffixes. They're either attached to roots with no suffixes at all or derivatives which only contain other derivationals.
"Bisikletlerimizsiz" doesn't comply with the rule. "-siz" is a derivational and there are a couple of inflectionals before it.
So, in cases you cannot use the suffix "-siz", as above, you resort to "olmadan" instead. You say, "Bisikletlerimiz olmadan".
Hope that helps. It's a bit hard to elaborate on things all the while trying to keep it simple.