Do you mean actual bids in an auction, or do you mean Best Offers ("Make an offer")?
If you mean actual bids, there is no "maximum" but things (e.g. eBay set Buyer Activity or Open Transaction Limits, seller blocks) can cause you to no longer be able to bid or buy from that seller or in general until you fall back below the limits. A way to minimize this is to not have many "Open Transactions" at any one time: don't bid until the auction is almost over then bid just ONCE at your TRUE maximum you are willing and able to pay and then if you win pay promptly, and go easy on the number of Best Offers you have pending at any one time.
If you mean Best Offers, the limit (for most categories) is 5 per user per listing. If you get to the 5th one, make sure it is at your TRUE maximum because you don't get another. Because you CANNOT make another on that listing, including any autorenewal of that listing (the "Time remaining" on a fixed-price listing does not mean it will end at that time, it means it autorenews for another month unless sold or the seller ends it manually). You can ask the seller to lower the Buy It Now price or end and relist it (so it gets a new Item Number and is a new listing with a blank slate on Best Offers) but most would be very wary of a buyer proposing that.
