You can have up to three potion bottles. The only way to get them is to advance through the main quest series (in other words, keep doing the quests in your book). Ambrose gives you the first one, I believe after you finish Unicorn Way. The second is given by Major Talbot in Krokotopia when you have earned the Key to travel there on your own. The third is given when you earn the key to Marleybone.
If you are running out of mana too often, look for gear that increases your maximum mana, so each of your potion bottles will hold more at one time.
Currently three is the max you can have. You are given them as 'gifts' for completing certain portions of the game. You get the first fairly early on in Wizard City, the second you get upon entering Krok, and the third upon entering Marleybone.