This is a good question and got me thinking - my admittedly long-winded thoughts below.
There are all sorts of factors that could be considered when determining the number of shares to hold at any one time. I don't set any limits - the number of shares that I hold is determined by my position sizing strategy.
Factors to consider;
1. Position sizing strategy. If you are using % risk position sizing strategy then you are not focused on the number of positions. You look at the amount of risk in each position. I know someone will raise portfolio heat as an issue, but I don't consider it as a major factor.
If you use 10% of capital in each trade then obviously the maximum number of trades will be around 10 - unless you add more capital to the mix.
2. The amount of trading capital. Someone with $10 million will probably hold more shares than someone with only $100,000. Liquidity becomes a factor and can limit position size and therefore the amount of capital in each trade.
3. Liquidity. No point trying to buy $100,000 shares in a company that only turns over $20,000 a day. So position size would need to be decreased, or the trade could be discarded.
4. Time frame that you trade. I trade weekly a weekly time frame so I have a bit of time to handle a larger number of positions. Time frame can also impact on liquidity as it is possible to take a position over a number of days, or even weeks, with a weekly or monthly time frame.
5. Holding length of trades. A buy and hold strategy can obviously handle more stocks.
6. Broker quality - how easy is to exit a number of positions at the one time. On-line brokers means that the trader does all the work. Handling a large number of positions in a short time-frame means that errors could creep in. Sometimes I think that it would be nice to ring up a broker and give him my orders for the week, but then I think of the brokerage costs!
7. Trading system / decision making approach. If you have to do vast amounts of research before making a decision this could impact on the number of positions held. If you scan the market on the weekend and do a minimal amount of research of the signals (or no research on the sells) then it is possible that more positions can be handled.
8. Portfolio growth - as my portfolio (hopefully) grows, or I add in extra capital, it takes time for the increased position size to filter through the entire portfolio. So I end up taking up more positions to soak up available capital. If I go to mainly to cash then I can start from scratch, but I can end up holding positions based on my capital from over a year ago, even though I more than doubled my capital.
I am sure that there are other factors I have missed, but these are the obvious ones. I currently hold 17 positions in my super fund With enough money to take 1 or 2 more positions. I have a similar number in my investment company, although it is only a 3rd of the size of my super fund.