[Ben Cohen]

So as you go down the halogens, the solubilities of AgCl, AgBr and AgI decrease. Is it because the bond Ag-X becomes less polar and thus, as with acids, the less polar a bond the less breakable it is and the less will dissolve in water?

Thanks in advance.

[Schrödinger]

Yes. As you move down the group, the bond's covalent character increases, which invariably is related to the decrease in difference between the electronegativities of the bonding atoms, which means decrease in polarity of the molecule. And, since you're talking about water here, less polarity (less ionic character) implies less solubility

[souro10]

@Ben

What do Fajan's Rules for covalent character say? Apply the same.

Note that, the solubility of silver halides in water is AgCl > AgBr > AgI due to increasing covalent character.

However, it would be a blunder to apply Fajan's Rules to the Sodium halides , lithium halides or other Grp 1 halides. The solubility of sodium halides in water is just the opposite of the silver halides with solubility increasing as follows - NaI > NaCl > NaBr ! Now, why do Fajan's rules predict the wrong order in this case?

Try to understand the chemistry involved in each case. Solubility does not depend on difference in electronegativity or polarity in all cases. In case a compound is predominantly ionic ( it has much more ionic character in its bond than covalent character ) , like Sodium Halides, covalent nature being minority is not the decisive factor for solubility. For such compounds, you need to look into how strong the ionic bond is. You know, larger the interionic distance, weaker is the ionic attraction , and consequently weaker is the (predominantly) ionic bond, and hence its easier for water molecules to break the bond and hydrate the ions - assuming all other factors to be favorable (read - hydration energies ). In case the bond has more covalent character than ionic character, then you need to apply Fajan's rules / the concept of polarization of ions i.e distortion of electron cloud of the anion by the cation.

So ultimately what predicting solubility boils down is to decide first whether the compound is predominantly ionic or predominantly covalent. After that, apply concept of electrostatic force or Fajan's rules accordingly. Now, how do you predict whether a compound will be predominantly ionic or predominantly covalent ?

Well.. there is no single , simple theory for predicting that. However, an useful tool is to use Pearson's Hard-Soft-Acid-Base theory. The theory is not successful in all cases, and one must note the exceptions/contradictions while studying it. Briefly stated, Pearson classified electrophiles as hard or soft, and nucleophiles as hard or soft as-well. Hard electrophiles/nucleophiles are those that are small , and have a high  positive/negative charge density and are highly electropositive/electronegative.  ( examples - Na+ , Li+ , H+ , F- , Cl -  ) Soft electrophiles/ nucleophiles are those that are large , have a relatively low positive/negative charge density and are not very electropositive/electronegative ( Ag+ , Pb2+ , I- etc )
Pearson stated , Hard electrophile prefers to react with hard nucleophiles to form ionic bond and soft electrophile prefers to interact with soft nucleophile to form covalent bond.  From this, its clear that NaF is highly ionic and AgI is highly covalent. But what about NaI and AgF ? The two cases are hard-soft and soft-hard interactions respectively. Pearson did not say anything about this in his HSAB theory. However, the theory can be extended to include these species. If the cation is very hard, the anion has no power to influence the cation to form a covalent bond in most cases, due to its large size ( exception - all nitrates including silver nitrate is soluble in water! But the reason is different. Nitrates are soluble due to highly favored hydration of nitrate ions - hydrogen bonding ) and a Very Hard(Electrophile) + Any Nucleophile => Ionic bond , mostly. However, if the electrohpile is borderline hard, or soft , while the anion is extremely hard, things can get reversed. Consequently, AgF is soluble in water.

Try to apply HSAB theory to as many compounds as possible as an exercise , and check whether you are right by checking solubility values from the internet! Each time you get an exception , try to frame a logic for why it didn't work. This way, you can gradually remember many of the simply compounds' solubility and exceptional cases, if you work them out yourself.