The short answer is "no" - you can buy a single share of any publicly traded company if you want to. Thus, if you have a small amount of money to invest, you can, in fact, buy a small number of shares of a public company. Most brokers will process a trade for a few shares of common stock, as they receive a commission for their services anyway.
However, just because you can invest your savings this way, does not mean that such an investment will be a good one. You have two major obstacles which must be mitigated before going ahead and buying a small number of common shares: diversification and transaction costs.
First, you need to strongly consider the costs of under-diversification if you are going to begin your investment portfolio with a single stock. If you have no other investments, investing in only one company exposes you to an excessive amount of company-specific risk (i.e. if your chosen company is the next Enron, you could lose virtually everything). Thus, if you have a small amount of money to invest, a much more efficient portfolio can be constructed by buying into a mutual fund. A mutual fund is essentially a large basket of investments bundled together, and will provide growth opportunities for a reasonably low amount of risk. (For further reading, see The Importance Of Diversification and Portfolio Protection In Diversification And Discipline.)
Second, even if you can stomach the risks of under-diversification, your next hurdle is a high one: transaction costs. Suppose your brokerage charges you $30 commissions for each trade. If you plan to buy and (hopefully) sell a stock for a profit, you will incur $60 of transaction costs. If you only had $200 to invest, your investment would need to gain 30% ($60/$200) simply to break even - an extremely inefficient investment. Conversely, if you invested the same $200 in an open-ended mutual fund, you would likely only be charged a small management fee of, for example, 3%. This would leave you with 27% of the 30% you would have had to spend on a single stock purchase. (To learn more, see Don't Let Brokerage Fees Undermine Your Returns.)
When combined, transaction costs and the risk of under-diversification usually prove to be too costly for those who only have a small amount of money to invest. Therefore, rather than purchasing a few shares of a public company, buying mutual funds is often a much better option.
To learn more, read Mutual Fund Basics Tutorial and The Advantages Of Mutual Funds.