Are you asking about "bids" on the auction, or "offers" through the Best Offer process? See below for more on the differences.
If you are asking about actual "bids", anytime there is only one bidder (regardless of how many bids s/he has placed) the "current bid" amount cannot rise above the Starting Price you set (unless you also set, and paid a substantial fee for, a Reserve Price which the bidder has met) because the price is set by the high bid of the SECOND highest bidder and eBay hides the full amount of any bid that exceeds that calculated price ("current bid" while the auction is running, which is simply what the price would be if the auction ends with no further activity).
If you mean an "offer", some sellers set the Starting Price at their absolute minimum sales price while others (especially new sellers) will set it higher (and some set it lower and wonder why they have to sell for less than they are willing to). It would not be unreasonable (but nothing I'd ever considered before) for a buyer to offer the Starting Price to provoke a jump-on-it-now reaction from ones who set it higher or a counteroffer from those who set it lower.
Too many people confuse "bid" and "Best Offer." Those are two entirely different things that the seller handles differently. People who come here and use the wrong term (what does the message or display ACTUALLY say you got?) often get the wrong answer (actually it's the right answer for the question asked but not what s/he needed to know).
A bid is part of an auction. Sellers don't "accept" them (eBay accepts them if they are valid bids: meet the minimum bid requirement as of the time they are received, received before the end of the last second of the auction, buyer is not prohibited from bidding on that auction because of a restriction by the seller or eBay). A seller can cancel any bid up until the auction ends (or eBay might if the buyer is suspended before the auction ends or the bidder might retract the bid for a very short very specific list of valid reasons) but otherwise the bid remains active until the auction ends, at which time a winner and price is set from the bids that were accepted by eBay and not cancelled/retracted before the end. The seller is obligated to sell for that price (if the buyer pays, provides valid address that seller agreed to ship to etc.) and the buyer is obligated to buy.
A Best Offer is simply an offer that the seller can take or leave. S/he can accept it, reject it, make a counteroffer (which acts as a rejection of the offer), or do nothing and the offer expires (at the duration the buyer chose when making it up to 48 hours, or earlier if there is a "bid" placed--Best Offer is no longer available after the first bid on the auction, or when the listing ends for whatever reason). The seller is under no obligation to accept any offer or respond in any way to a particular offer or any offer at all.
Inceptions
